Why does the Kenmore washing machine not move out of the wash cycle

You have a crack somewhere in the drainage system. The most likely reason that it's leaking on the spin cycle is the vibration caused by the spin cycle is opening the crack to let the leaking occur.

(Summarizing the comments above)

To narrow this problem down, there are three places that could be causing the GFCI to trip, a malfunction in the washing machine, a problem with the downstream wiring (aka load side of the GFCI), or the GFCI outlet itself. If there isn't anything downstream, then plugging the washing machine into another GFCI outlet, or simply swapping out the outlet for a known good GFCI outlet, will identify if the outlet itself is faulty.

If the outlet trips when the washing machine isn't running and isn't even plugged in, then there's a fault in the wiring on the load side of the GFCI outlet.

If the issue is neither of the above, then running the washing machine and monitoring to see which step is occurring when the trip happens will isolate what part of the washing machine may be leaking current to a ground. It could be a certain water level, a motor being engaged, a transition step in the controller, etc.