No electricity to fuel pump, what are most likely problems? fuse is good, checked fuel pump relay, when key is on, only one wire of 5 is hot.

i see you are guessing?
did you check all engine related fuses? that be first.
with a $20 DMM volt meter , the guessing ends.
the electrics form 2 paths,
1: power
2: ground
if either are bad, the pump fails. (suzuki love to do this)

when you turn the key, the ECU/PCM , energizes
the fuel pump relay,
if you have no power to the relay the relay will be dead.
the relay then passes pump power to the pump, on the pink wire
on most suzuki's state side (no location stated)
the pink wires goes hot (12v) for 3 seconds each key on.
if not the relay is bad, or the power to it is bad or the ECU is bad.
if you lose ground to the relay , the ecu or the pump. the pump
will be dead.
using a voltmeter allows you to find this simple failure.
lets pretend the pink wire is hot.
you key on, pink goes 12v , for 3seconds but the pump is silent.
1: the pump has 2 wires , not 1.
one or the other is open.
2: if you see 12v on the pumps ground wire for 3 seconds key on.that means the pump ground is dead open.
the voltmeter will not lie to you , use it and find the problem easy.

If you have no fuel pressure at the engine when key is on, the pump is not coming on. Check fuel pump fuse, check fuel pump relay (relay should click when key turned to on-put finger lightly on it and feel for click when someone turns key to on). If no click, signal from computer may be lost, ground for relay may be lost, or relay may be bad. If it clicks, wiring from relay to pump may have failed, or the relay's main power source may not be there at relay-find the fuse link or maxi-fuse that powers that relay. One terminal at relay should be hot at all times-can check for that with a test light.
If pump is working and you do have fuel pressure, you will need to check injector circuit. Post back if questions, or need further help.

Check the fuse for the pump first. You can check that fuse for power when key is on with a test light. Then find the fuel pump relay, probably in relay box under hood, and put your finger lightly on it and listen and feel for it to click when someone turns the key to on.
If relay does click, the problem is likely with the wire to tank from relay, the connector at the tank, or the wire going into the pump.
If relay doesn't click, or if it does and you haven't found problem yet, pull the relay out and with test light grounded to engine, probe all relay terminals for one hot terminal- one will be hot at all times. If no hot terminal, could be fuse link or maxi-fuse that feeds that relay box. If one terminal is hot, have someone be cranking the engine over while you check the other terminals for a second hot terminal. When cranking, two terminals should show power, the main feed for pump, and the signal from computer that energizes relay so power is passed to pump. If two terminals are hot when cranking, try a known good relay. If still no help, problem is with wiring to the pump from relay, or the ground for that relay needs checked.
Post back with what you find. Good luck.

There would be a fuel pump fuse, either under dash or under hood. Check that fuse, use the test light to see if fuse has power. If not, then it might be a fusible link or maxi-fuse that feeds the fuse panel. If fuse has power with key on, pull the relay, and use the test light and probe relay terminals, key off. One terminal should be hot at all times. If not, check for power loss between battery and relay terminal-possibly a maxi-fuse or fuse link.
If relay does have power to terminal, quickly check other terminals while someone turns key to on-for about 2 seconds, another terminal should show power-this would be signal from computer to turn the relay on to prime the lines and fuel rail. If you have that, plug relay in and cycle key. Does relay click? If yes, relay is working and problem may be at fuel pump connector. If relay is not working, but you have power to it, and the short signal from computer is there, then check the relay's coil side ground. Computer signals the coil side and the signal goes to ground from relay coil side.
Pull the relay and use a jumper wire from relay's hot terminal to fuel pump terminal-the wire to pump from the relay. If pump comes on, the problem is right there at the relay-either no signal from computer or the relay coil's ground is lost. If pump does not come on, again check connector at pump-that or the wire to pump[ has a problem.

The power to the pump comes through the relay.
To verify the pump circuit is working right, and that the fuel pump needs replacing, do the following test: You will need a voltmeter or test light, and a helper. The pump wire at the tank is a white wire with, it appears, a blue stripe.
This may have been the wire you saw a brief signal on. In fact, it should have a 2 second signal when key is first turned to on-this is to prime the lines and system for starting.
There should be a steady signal on this wire when the engine is being cranked over. The relay should turn the pump on full time during cranking and running. So have a helper crank the engine over while you check for power on the wire. If there is power then, the pump circuit is working right and you probably have a failed fuel pump motor. The only other possibility is a bad ground wire for the pump-which you probably should also check.
You are right about the relay having two hot wires or terminals with key on, so that much is correct. The ecu turns the relay on when it grounds the coil side of the relay.

Your fuel pump is powered by the GREEN fuse link in the fuse link holder. If you have 2 terminals (both WHITE/BLACK wires) at the fuel pump relay that is HOT all the time (even with the key OFF) then this fuse link is OK. There are no other fuses to be concerned with.

There is also a "Fuel Pump Control Unit" located on the left rear quarter panel, behind the "B" pillar. (See Picture)

I hope this helps? Are you missing anything else? Engine cranking you have spark at the plugs and fuel injector pulse? Any applicable trouble codes? Battery has a full charge? See load side of mfi relay, that line with arrow head, with key on, good voltage at upper wire? If so, if you use a jumper wire between proper wire colors, the pump should run.

The wire should not be hot until the relay is triggered. The relay does not get triggered until the ignition key is turned on. It will run for 3-5 seconds or until the system pressure it at max.

Look under hood some ware around left fender well there should be a red wire with a plug on it that looks like it should be pluged to something but its not, that wire goes directly to fuel pump, run a hot lead to that and see if pump runs. Also chk. grounds and hot leads and all fuses and the relay. If you can't find the wire, and your sure fuses and relay and connections are functioning, you may have to run your own hot wire to pump to see if it will run. Be aware its possible to have gotten a defective pump.

you should have power on the tan wire for about 5 seconds with key on try to cycle key on for 5 sec off for 10 sec if no power chehave power on both sides of fuseck ecm 1 10 amp fuse if you check ecm b fuse if that is ok check for power on the orange wire and power on the green with white wire on fuel pump relay if you have power on those check for power on the grey wire out to fuel pump if you have power on grey wire you have a bad wire from the grey to the tan wire some where in the harness hope this helps