1986 rx-7 has overheated a couple times and now it will not start-it will only start after a push start-there is no starter noise what could be wrong?

I would first check for voltage at the ignition switch start feed wire on the solenoid(be sure to unhook it first from solenoid). If you are getting power on this wire when turning to start. It should be an issue with the solenoid/starter. Some part of this assembly could have been damaged from the overheat. If no power to the crank wire, I would get a hold of a repair manual and trace the crank wire from the ignition switch all the way to the starter, making sure it has battery power to that terminal on the solenoid. Once you have that and a known good starter she should crank right up. hope this helps.