Cooling fan doesn't come on when A/C compressor is engaged. Relay is okay. No ground for relay coil.

Possible bad fuse, relay or switch.

It means you should take your vehicle to a qualified repair shop , at least that's the car makers thought .
A/C Off For Engine Protection
Refer to Cooling System Description and Operation
Cooling Fan Control
The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the powertrain control module (PCM) or engine control module (ECM) to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood junction block. The ground path is provided at G100.
During low speed operation, the PCM/ECM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the cooling fan 1 relay coil, closes the relay contacts, and supplies battery positive voltage from the cool fan 1 maxifuse?® through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the cooling fan relay and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation, the PCM/ECM supplies the ground path for the cooling fan 1 relay through the low speed cooling fan relay control circuit. After a 3 second delay, the PCM/ECM supplies a ground path for the cooling fan 2 relay and the cooling fan relay through the high speed cooling fan relay control circuit. This energizes the cooling fan relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the cooling fan 2 relay coil is energized closing the relay contacts and provides battery positive voltage from the cool fan 2 maxifuse?® on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.
Are the radiator fans coming on ? You mite take it an have it checked for DTC'S - diagnostic trouble code's .

Why ? Did you test ? Hook up a scan tool to see if the PCM/ECM - engine computer is reading coolant temp. ? Testing not guessing ? Do you even know how the system works .

The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the engine control module (ECM) to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood fuse block.
During low speed operation, the ECM supplies the ground path for the low speed fan relay, which is Fan 1 Relay, through the low speed cooling fan relay control circuit. This energizes the low speed fan relay coil, closes the relay contacts, and supplies battery positive voltage from Fan 1 Fuse 29 through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the Series/Parallel relay, which is the Fan 2 Relay, and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation the ECM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. The ECM supplies a ground path for the high speed fan relay, which is the Fan 3 Relay, and the Series/Parallel through the high speed cooling fan relay control circuit. This energizes the Series/Parallel relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the Fan 3 Relay coil is energized, the relay contacts close to provide battery positive voltage from Fan 2 Fuse 33 on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have their own ground path. The result is a parallel circuit with both fans running at high speed.
When the request for fan activation is withdrawn, the fan may not turn OFF until the ignition switch is moved to the OFF position or the vehicle speed exceeds approximately 10 mph. This is to prevent a fan from cycling ON and OFF excessively at idle.

A/C Off For Engine Protection
Refer to Cooling System Description and Operation

DTC P1258: Engine Coolant Overtemperature - Protection Mode Active.
This is something you need to take to a qualified repair shop .
The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the powertrain control module (PCM) or engine control module (ECM) to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood junction block. The ground path is provided at G100.
During low speed operation, the PCM/ECM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the cooling fan 1 relay coil, closes the relay contacts, and supplies battery positive voltage from the cool fan 1 maxifuse® through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the cooling fan relay and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation, the PCM/ECM supplies the ground path for the cooling fan 1 relay through the low speed cooling fan relay control circuit. After a 3 second delay, the PCM/ECM supplies a ground path for the cooling fan 2 relay and the cooling fan relay through the high speed cooling fan relay control circuit. This energizes the cooling fan relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the cooling fan 2 relay coil is energized closing the relay contacts and provides battery positive voltage from the cool fan 2 maxifuse® on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.

The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the powertrain control module (PCM) to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood fuse block.
During low speed operation, the PCM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the low speed fan relay coil, closes the relay contacts, and supplies battery positive voltage from the low fan fuse through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the cooling fan s/p relay and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation the PCM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. After a 3 second delay, the PCM supplies a ground path for the high speed fan relay and the cooling fan s/p relay through the high speed cooling fan relay control circuit. This energizes the cooling fan s/p relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the high speed fan relay coil is energized closing the relay contacts and provides battery positive voltage from the high fan fuse on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.
Your best bet would be to take your vehicle to a qualified repair shop !
Relay Driven Dual Cooling Fan Diagnostics

there is 2 fuses that you have to check
one will be in the big fuse assembly in the engine bay, the other probably in the fuse assembly in the cab
If they prove ok then it is possible that the coolant temperature SENSOR is faulty and needs replacing
don't confuse it with the temperature sender which operates the temp gauge.

The engine cooling fan system is controlled by the body control module (BCM) and the powertrain control module (PCM) or engine control module (ECM). The BCM performs the calculations as to how long, when and what speed the cooling fans should turn on. The BCM then sends a class 2 message to the PCM/ECM to engage the cooling fan relays. If there is a malfunction with the BCM, the PCM/ECM will control the engine cooling fans independently. The engine cooling system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the PCM/ECM to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood accessory wiring junction block. The ground path is provided at G100.
During low speed operation, the PCM/ECM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the cooling fan 1 relay coil, closes the relay contacts, and supplies battery positive voltage from the cool fan 1 fuse through the cooling fan motor supply voltage circuit to the left cooling fan. The ground path for the left cooling fan is through the cooling fan 2 relay and the right cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation the PCM/ECM supplies the ground path for the cooling fan 1 relay through the low speed cooling fan relay control circuit. After a 3 second delay, the PCM/ECM supplies a ground path for the cooling fan 3 or S/P relay and the cooling fan 2 relay through the high speed cooling fan relay control circuit. This energizes the cooling fan 2 relay coil, closes the relay contacts, and provides a ground path for the left cooling fan. At the same time the cooling fan 2 relay coil is energized closing the relay contacts and provides battery positive voltage from the cool fan 2 fuse on the cooling fan motor supply voltage circuit to the right cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.
The BCM and PCM/ECM command Low Speed Fans on under the following conditions:
?€¢
Engine coolant temperature exceeds approximately 106?°C (223?°F).

?€¢
A/C refrigerant pressure exceeds 1 310 kPa (190 psi).

?€¢
After the vehicle is shut off if the engine coolant temperature at key-off is greater than 140?°C (284?°F) and system voltage is more than 12 volts. The fans will stay on for approximately 3 minutes.

The BCM and PCM/ECM command High Speed Fans on under the following conditions:
?€¢
Engine coolant temperature reaches 110?°C (230?°F).

?€¢
A/C refrigerant pressure exceeds 1 655 kPa (240 psi).

?€¢
When certain DTCs set.

The three relays are located in the under hood fuse / relay box ! You need to have it checked for DTC'S -diagnostic trouble codes ! An you need a GM Tech 2 scan tool !
This diagnostic procedure supports the following DTCs:
?€¢
DTC P0480 Cooling Fan 1 Relay Control Circuit

?€¢
DTC P0481 Cooling Fan 2 and 3 or S/P Relay Control Circuit

?€¢
DTC P0691 Cooling Fan 1 Relay Control Circuit Low Voltage

?€¢
DTC P0692 Cooling Fan 1 Relay Control Circuit High Voltage

?€¢
DTC P0693 Cooling Fan 2 and 3 or S/P Relay Control Circuit Low Voltage

?€¢
DTC P0694 Cooling Fan 2 and 3 or S/P Relay Control Circuit High Voltage

1. Install a scan tool.
2. Turn ON the ignition, with the engine OFF.
3. With a scan tool, command the Fans Low Speed ON and OFF.

Do the low speed engine cooling fans turn ON and OFF with each command?
Go to Step 3
Go to Step 4

3

Important:: A 3 second delay occurs before the powertrain control module (PCM)/engine control module (ECM) changes the cooling fan speed.
With a scan tool, command the Fans High Speed ON and OFF.
Do the high speed engine cooling fans turn ON and OFF with each command?

The engine cooling fan system consists of 2 electrical cooling fans and 3 fan relays. The relays are arranged in a series/parallel configuration that allows the powertrain control module (PCM) to operate both fans together at low or high speeds. The cooling fans and fan relays receive battery positive voltage from the underhood accessory wiring junction block. The ground path is provided at G113. The relays are located in the underhood fuse / relay box .
So ,what's the problem Paul , car overheating ? Fans not working ?
During low speed operation, the PCM supplies the ground path for the low speed fan relay through the low speed cooling fan relay control circuit. This energizes the cooling fan 1 relay coil, closes the relay contacts, and supplies battery positive voltage from the cool fan 1 fuse through the cooling fan motor supply voltage circuit to the right cooling fan. The ground path for the right cooling fan is through the cooling fan 2 relay and the left cooling fan. The result is a series circuit with both fans running at low speed.
During high speed operation the PCM supplies the ground path for the cooling fan 1 relay through the low speed cooling fan relay control circuit. After a 3-second delay, the PCM supplies a ground path for the cooling fan 2 relay and the cooling fan 3 relay through the high speed cooling fan relay control circuit. This energizes the cooling fan 2 relay coil, closes the relay contacts, and provides a ground path for the right cooling fan. At the same time the cooling fan 3 relay coil is energized closing the relay contacts and provides battery positive voltage from the cool fan 2 fuse on the cooling fan motor supply voltage circuit to the left cooling fan. During high speed fan operation, both engine cooling fans have there own ground path. The result is a parallel circuit with both fans running at high speed.
The PCM commands Low Speed Fans on under the following conditions:
?€¢
Engine coolant temperature exceeds approximately 106?°C (223?°F).

?€¢
When the A/C is requested and the ambient temperature is more than 50?°C (122?°F).

?€¢
A/C refrigerant pressure exceeds 1310 kPa (190 psi).

?€¢
After the vehicle is shut off if the engine coolant temperature at key-off is greater than 140?°C (284?°F) and system voltage is more than 12 volts. The fans will stay on for approximately 3 minutes.

The PCM commands High Speed Fans on under the following conditions:
?€¢
Engine coolant temperature reaches 110?°C (230?°F).

?€¢
A/C refrigerant pressure exceeds 1655 kPa (240 psi).

?€¢
When certain DTCs set.

So

Are you talking about the radiator fan? How hot does the engine get in degrees F? Some makes and models, the radiator fan won't turn on till about 220 degrees F. Check for applicable trouble codes, see if any code that refers to the radiator fan circuit or coolant temp sensor? Some of the national brand auto stores will check codes for free.

Of course, as far as overheating in general, you have the thermostat--water pump and other possible issues. Are you losing coolant?

I would test the A/C compressor clutch coil first , easiest to do . If the electromagnet clutch coil is bad ,the compressor won't engage . Pull the A/C compressor clutch relay an jump pin's 30 & 87 , where the relay plug's into the fuse box . If pin 30 has B+ voltage ,pin 87 sends that B+ voltage to energize the clutch coil . These clutch coil's can go bad . You could also check the cooling fan this way , pull the cooling fan relay an take a piece of wire an jump pin's 30 & 87 .If fan is good it will run . Do you know how relay's work ?
VEHICLE RELAYS Operation Diagnosis

The PCM controls both relay's by supplying a ground to energize the relay .
Is the check engine light on ?
Do you know what a wiring diagram is ? How to use one along with a DVOM - digital volt ohm meter to test automotive electrical circuits ? Not guessing ? This is how an automotive technician would solve this problem , hooking up a scan tool an checking for code's ,using bi-direction control of scan tool to manually try to turn on coolant fan ,A/C compressor .....
Free wiring diagrams here http://www.bbbind.com/free_tsb.html
Enter vehicle info. year , make , model an engine size. Under system click on engine ,then under subsystem click on cooling fans ,check the electrical circuit's .