I like to check-raise in spot like this. If the villain raises and all the chips goes middle, that´s fine. If our opponent folds, that´s fine too. You are a favorite against a hand as strong as QQ.

I would 4bet get it in pre flop wth AK OOP. It does look like their bet range will be pretty strong but theres still plenty of spewy plays in their range.

A played I don't thin you can go too wrong with eiher XR, XC or even donking the flop we have so much equity against most ranges. Even against a range of AA, KK and QQ w are still ahead so I would be raising to happily get it in but also to fold out any other AK hands we tie with or random bluffs (like SCs)that we can get to fold equity.

I'd prefer a 4bet preflop. Once this flop hits you're a favourite against hands as strong as KK and this hand also benefits lots from having fold equity since it's a semibluff so I would be raising flop. I'd make the raise fairly small so I can make a large bet on the turn but I'm not sure if it matters much considering money will likely go in on the flop with a lot of villain's continuing range.