Penalty on Offense - Clock run?

[Spence]

I'm not an official. Someone asked me about the following situation.

Late in the game. Offense winning. Right before the play clock was to expire the offense was flagged for a false start. He says that the official set the ball and then started the clock. Offense did the same thing again.

He wants to know if in high school the clock should not start until the snap on a penalty against the offense.

Thanks