Difficulty starting - '90 80

[Robert Rossato]

Looking for some help for a friend of mine.  He has a 1990 80q.  He was
driving around this afternoon and after a stop for a few minutes the car
would not start, or more accurately, would not keep running.  Previously
he had been driving around for 20-25 minutes.

The engine starts immediately but after 5-6 seconds it will shut off.
He can start it back up immediately, but again after 5-6 seconds the
engine shuts off.  Very consistent, though on a couple of occasions the
engine ran for about a minute, during which he could freely rev it up
and down, but then shut off again.

When it shuts off it feels like it's running out of fuel, rather than a
hard ignition shut off.  Tach slowly goes down as engine starts to miss,
stumble, and finally die.  No amount of throttle will keep it running.

Given it starts immediately and runs for a few seconds, and on a couple
of occasions for a minute or so, I don't imagine it is ignition or valve
timing related.  

Fuel pump is audible.  

It's been raining hard all day here in Connecticut, but engine
compartment looked dry, he purposely avoided deep puddles and the car
had been running for 20+ minutes before he stopped to run an errand.  I
suppose moisture could be a contributor but why does it start
immediately?  I wouldn't expect that from wet ignition or the like.

Also, I would imagine a vacuum leak or rupture of an intake boot would
not let it run for 5-6 seconds.  In my experience an intake leak would
start and shut off almost immediately.

It sounds like it's running on the fuel from the cold start system.  He
tried starting the car off and on for over an hour while waiting for a
ride home (me).

Any thoughts?

Fuel pump relay?
Weak/overheating fuel pump? It's been getting loud lately, but why would
it start back immediately after it shuts off?
Any sensors that could be causing this?

Thanks in advance.

Bob