2005 GMC SLT ENVOY rear washer wiper blade not working.fuse okay

I wonder if the Immobilizer is related to the door locks and the fuse blowing broke the pairing. Try the reset

https://www.google.com/search?q=2005+GMC+Envoy+SLT+reset+security

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Did you check the fuse ? Do you hear the wiper motor running ? If the fuse is good you would need to test the electrical circuit for the wiper motor ,viewing a wiring diagram an then use a DVOM - digital volt ohmmeter to test
What body type is it ? Vin 6 or 3 could have a circuit breaker for power supply or fuse ,both are located in the rear fuse block .

Rear Wiper/Washer System Components
The rear wiper/washer system consists of the following components:
• Rear window wiper motor
• Rear window washer fluid pump
• Rear wash relay
• Rear window wiper/washer switch
• RR wiper 15A circuit breaker
• WASH 20A fuse
Rear Wiper/Washer System Controls
• Body control module (BCM)
• Liftgate control module (LGM)
• Rear wiper/washer switch

Here is how it is controlled

Rear Wiper/Washer System Operation
The rear window wiper/washer switch is an input to the BCM. The requested rear wiper mode is sent to the liftgate control module (LGM) by the BCM as a message on the serial data circuit, and the rear window wiper motor is controlled by the LGM. The switch power circuit to the rear wiper/washer switch is a 12-volt reference supplied by the BCM during ignition on, accessory or RAP power modes. The rear wiper/washer switch is a series of resistors and the switch position is determined by the voltage of the rear wiper/washer switch signal circuit to the BCM. The rear wiper/washer switch has continuity at all times and the BCM expects a rear wiper/washer switch signal voltage in all switch positions including OFF. The rear wiper motor is supplied battery positive voltage and ground. The LGM controls rear wiper motor operation through the rear wiper motor control circuit. The rear wiper motor control circuit is supplied a battery positive reference voltage by the rear wiper motor module and is pulse width modulated to ground by the LGM. The duty cycle of the pulse width modulation determines the rear wiper motor operating mode. The liftglass and liftgate ajar switches are inputs to the LGM and must be in an inactive state in order for rear wiper washer system operation to be enabled. When an ajar switch is active the LGM suspends wiper motor operation directly and sends the ajar switch status messages over the serial data circuit enabling the BCM to suspend washer pump operation. The rear window washer pump is controlled by the BCM through the rear washer relay. Battery positive voltage is supplied to the coil and switch sides of the rear washer relay. When the rear window wash request is active the BCM energizes the relay by grounding the relay coil through the rear washer relay control circuit. When the relay is energized battery positive voltage to the relay switch is supplied to the washer pump control circuit.

Your best bet mite be to have a qualified repair shop check for codes in the BCM or LGM
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DTC B1017 or B3970

The rear wiper motor is supplied battery positive voltage and ground. The liftgate control module controls rear wiper motor operation through the rear wiper motor control circuit. The rear wiper motor control circuit is supplied a battery positive reference voltage by the rear wiper motor module and is pulse width modulated to ground by the liftgate control module. The duty cycle of the pulse width modulation determines the rear wiper motor operating mode.

There is also a relay ! FOG LP fuse # 11 10 amp location , under hood in the engine bay fuse / relay box . FOG LP relay #39 .

RR FOG LP relay #45 , RR FOG fuse #5 10 amp , location rear fuse box .
The fog lamp relays are computer controlled , by BCM

It sounds like you have a grounding issue somewhere. Look up a fuse diagram for the cabin fuse panel. Start from there.

It could be an electrical problem or a mechanical problem in transmission or transfer case. does it shift into 4 wheel high okay

Replace the blade and the blade holder. These days everything but the arm gets changed....and gone are the days of just replacing the rubber...

{: ) Check first the WPR fuse 20amp at the underhood fuseblock. If okay, check the operation of the REARWPR relay. You should hear or feel a click when you turn on the rear wiper switch. Or you can try swapping it with another similar relay that is known to be working. If not successful, then a deeper test should be performed like voltage and signal tests, testing for open or shorted components and the wirings

First thing to do is check the fuses. Check ac fuses and relays and auxilary relays. If okay you may have a bad blower motor or blower motor resistor

door lock actuator solenoid ( inside of door) needs to be replaced

you have to remove the air filter box to get to the washer fluid resivoir the pump is mouted to it