No water enters on rinse cycle

Without Model number I am only guessing but I suspect you have a blocked sensing tube that travels from the Level selector switch to the side of the tub somewhere. It is possible for this plastic tube to become partially clogged and not allow trapped air to vent when tub is drained of water. Usually this problem manifests itself as an overflow of the tub. Check the full length of this clear hose and see if the end nearest the tub looks like it may have debris or dirty detergent clogging it.

This hose is pushed onto a nipple mounted to a reservoir on the side of the tub. Push a thin copper wire 4-5 inches into the those after removing from nipple and also down into the nipple an inch or two before reinstalling the hose to the nipple. You MIGHT even blow into the hose while you have it off and you SHOULD hear a clicking sound coming from the level selector switch body.

Good Luck. Come back with Model number under lid if that doesn't help.

So when the cycle first starts, it will fill and agitate and spin. But then on the rinse cycle it stops?

sounds like the solenoid to set the transmission into the Spin cycle is not functioning.

probably you've got something blocking the water pump from doing its job properly so check that the water pump is free running emptying all the water if not then there is a blockage in the pump normally ist a button, paper clip or other debris out of a trouser pocket eric

if no water is coming in how is it agitating?is it filling at the beginning,then it starts to wash the clothes,now does it drain and then it won't fill again,send the model number and more info,

Water level switch most likely. Seen it.

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you need to replace the water level switch

Please be carful as running the agitator without enough water will overheat and kick out the motor, however it should be ok when it resets itself.
Yes, the machine should stop agitating when you lift the lid so this too is part of the problem.
After looking at the print I can only suggest that the cam contacts in the timer ( contacts 12 and 14) that start the motor are burned and stuck together.
Quentin

while the unit is filling in the warm cycle. reach behind and shut off the hot water tap.I bet the water flow stops filling, if so, you would need to change the inlet valve on the machine. You see, in warm, if the cold is not filling, you still hear the hot filling and the unit will advance. The rinse cycles are mostly "cold" only. if the valve is not opening it will not allow it to advance from the rinse section on the board.

Good luck!

Check your timer and make sure that it is working correctly with a multi meter