[OverAndBack]

Oops. My bad.

Here's the relevant NFHS rules:

Rule 6, Section 5, Article 4:
"The captain may choose to free kick or snap anywhere between the inbounds lines on the yard line through the spot of the catch when a fair catch is made or through the spot of interference, when a fair catch is awarded. These choices remain if a dead-ball foul occurs prior to the down, or a foul or an inadvertent whistle occurs during the down and the down is replayed."

I can't find anything that says the half can or can't be extended by an untimed down in the instance of a fair catch as the clock hits zeros. It wouldn't surprise me if it was the case, though.

I knew you could fair catch and free kick. The timing issue I wasn't aware of.