1997 Runs Sometimes, sometimes not?

Daughter's '97 Sunfire w/2.4L, manual xmission, 120K, starts fine, runs about 45 seconds, stalls and won't restart for a couple of minutes, sometimes. Then starts, may or may not continue to run. New wires, new pwr pack and fuel pump (by prev owner) didn't cure problem. Does this in all kinds of weather, fair or foul, hot, cold, and everything in between. Once underway, operates fine for hours at a time. Check engine light isn't coming on so there are no fault codes to read. She says sometimes the fuel pump doesn't sound like it's cycling when the switch is turned on but I've failed to notice this. Pump relay is secure in its socket and contacts are clean. Only clue is that engine temp may not have quite reached midpoint but if anywhere near cold, this seems to be happening with more regularity. All other functions seem fine and car is otherwise a great runner and in excellent shape. Is there a fuel pump cutoff switch on this model? Had a couple of Ford Escorts that would get intermittent cutoff switches but once reset was pressed, never had another problem. Odd.
Thanks for any help you can render!
Michael1

Get a dealer to check the PasslockII system in the car and make sure it's working properly (could be a problem with the actual transponder in the key). If it's having issues, then normally after piss it off enough it will lock out for about 15 mins and cut fuel and/or ignition to the engine.