OK
Check your solenoid with a test light or meter. Check power on the battery side key on, hit starter switch look for any change. Check power on starter side key on, hit starter switch should have power on starter side same as battery side when starter switch is activated. If not, check power and ground at solenoid activation terminals. If they're good replace your solenoid. Like Benny says, take it to an auto parts store and match up with something. It could be that the charger is supplying enough power to overcome a solenoid that's not making good contact internally.
