When playing heads up reads are pretty important, but I'll give my 2c anyway as if I'm playing an unknown:

H1: c/r flop, lead turn/priced to call a shove probably if he's got JT/QT/AQ/AK or something like that in his range. He does NOT ONLY have AK here.

H2: What a strange spot...I probably just call, if he's got a stupid FD you want him to hit it, then value town him. Either way you probably won't gain more or less by calling or raising.

Edit: H1