Another KK with A on the flop hand

[Nathan183]

Read on villain is solid TAG, hasn't done anything crazy, a good player I think.

Don't have the hand history, so I'll do it from memory.

$50 NL. I think I had ~$70 and villain had me covered.

I'm in EP, UTG or UTG+1 and raise to $2 with KK. Villain min raises to $4 (this confused me, I thought he was a good player!). Folded back to me and I make it $9 (too small?). After a long pause, he calls.

At this point, I've shown a lot of strength, and don't think he's calling here with less than QQ (maybe JJ) or AK.

Flop is A82 rainbow. QQ seems like the most likely hand now, although AK and AA are still possibilities, and less likely JJ or a AQ.

I bet $10 with the intention of folding to a raise.

Is this a good move? I don't want to give QQ a chance to bet me off of this pot. I've clearly defined my hand with preflop action, and I think checking would basically be saying "I have KK, take this pot from me".

He calls the $10. This surpised me, as I expect a raise from AK. Is he trapping with AK or even AA?

Turn is a blank and I check. He pushes. Another strange move I think. Why would he just call the flop and push the turn? My thought was that he could actually have QQ and sensed weakness with my turn check and was trying to get me to fold KK. I thought for awhile, and decided not to give a $50 NL player this much credit. I folded. What did villain have?