CGT cooling fan problem?

[JOlson1967 at aol.com]

I looked at a '86 CGT that had this problem:

The cooling fan would not come on with the car running and it would 
subsequently overheat.  Neither would the fan come on when the a/c switch was 
turned on.

However, when the ignition was turned off the fan would turn on and would 
immediately stop if the ignition switch was turned back on.  

The fan itself works fine but there seems to be a problem in the ignition 
switch or a relay.  I assume the fan thermoswitch is functioning properly 
because the fan turns on when the ignition switch is turned off and the car 
is hot.  The fan runs for several minutes and then shuts off.

The a/c switch does not turn the cooling fan when the car is running or the 
ignition switch is turned on, however the a/c does produce cool air.  I 
assume that when the starter is engaged, a relay is supposed to interrupt the 
power to the cooling fan and then turn the fan back on when the ignition 
switch is returned to the on position. 

I could find no obvious electrical problems that would the fan not to turn on 
when the car is running and is hot.  Again, as soon as the ignition is turned 
off, the fan will turn on if the car is hot.

Speculation is either a fault in the ignition switch or a relay?  Perhaps the 
thermostat or thermoswitch is not functioning properly?  Also, why won't the 
fan come on when the a/c is switched on?  It should come on regardless of 
coolant temperature, correct?

Any input would be greatly appreciated!  Thanks!