Engine doesn't start

I have a Mastercraft Prostar and recently the engine does not start after switching it off so in practice it does not start when warmed up.
What happens is that the starter motor does not even turn when the ignition key is switched to the start position. The starter motor seems dead. This goes on for a while and then when the engine cools down suddenly the starter motor is still alive and the engine immediately starts.
I had the starter motor checked and everyting was fine. This happens only when the engine is warmed up because when it is cold I do not have any problem with the starter motor.
Any ideas?

How many volts are at the in and out of the starter soleniod? Holding the key in the crank position can you rap on the soleniod and get it to fire?

I do not know how many volts are there. I will have to measure. When I hold the key in the crank position and the engine is warm the starter motor does not operate at all. I do not even hear any "click" coming from the starter motor. It is completely silent and obviously the engine does not crank.
I then have to wait for a while (about 15 minutes) and suddenly everything works again.

Either you have a bad starter or you are not getting the correct voltage to the starter.

Yes, but the question is why it only happens when the engine is switched off and then restarted, that it when the engine is warm.

Because starters fail more when hot then cold.

How many soleniods does the engine have on it? Is one mounted directly on the starter?

Again test for voltage at all the different pins as someone has the key in the crank position.

Thanks Peter, I will inspect all the parts you mentioned.

At present I am not where my boat is located so I cannot tell if one solenoid is mounted directly on the starter. What I know is that the engine is an Indmar 351 year 1991.
Here in Italy service for those engines is very poor since we do not have many of them.

I would guess it is a faulty solenoid

Luca. Check all wiring leading to and from the starter. This problem can happen when they start to burn through the insulation.

Hit the starter a few times with a wrench while someone tries to start the boat. If it starts, you need a new starter or get yours rebuilt. If it doesn't, find a voltmeter and make sure all the correct signals are getting to the starter. (+12.5V constant on the fat red wire, +12.5V on the skinny wire when someone turns the key to the start position)

I bet there are local electrical shops in Italy that can rebuild it. You could also take the starter apart and clean the rust out and probably limp along until you can get a new one.

Hi Trace, the fact is I already had the starter rebuilt but the problem is still there. I do not know if who rebuilt the starter made a good job but it is his daily job.
May be the correct signals are not reaching the starter. I will test it with a voltmenter, or better, I will have it tested for me since I have no spare time at the moment.

Hi Rob, I think there is a great chance that you provided me the right solution, together with testing with a voltmeter. Thanks to all.