You messed a bit with the post, on turn it is probably button who bets as cutoff is your girlfriend.
I noticed that paired boards are quite difficult to play, I even going to note what happened on every paired board that I am going to see played to have some statistical reference. The problem is that a lot of players like to bluff at that board, and a lot of times when I tried to pick their bluffs with overpairs they had trips
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As to the hand, read dependent, but it is not push or fold, if he has a J he will call your push, if he has a small pair he will fold. So it is fold, or call him down.
I think I would call turn in most cases and try to decide what to do on the river.