12.5 volts is low for a heathy battery after turning off. Its normally about 13 volts after a few minutes and will drop to 12.7 overnight, any less and the batter is on its way out.

An old school alternator would deliver about 14.4 volts to the battery after startup dropping to about 14.2 volts after some running. New style "smart" alternators vary voltage with load thus can be more difficult to assess. No idea if your car has one.

Suggest you go to a good auto electrician to get the thing checked over.