Washer fills with water, timer advances but no agitate (ever). Gets to end of cycle with no noises, then pumps water out and spins clothes normally.
Also nothing happens during rinse cycle.
The educated guess here is a timer problem. Before ordering one check out the wiring and clean and re-seat the connections to the timer.
I would call in the order with both your model number and serial number tomake sure you get the right part.
You also could have a bad water level switch. I would pull the cover off the timer and check for burnt contacts. If so, order a new one. AP4026232