Can Forged activation order choose ice hosted on Awakening center?

Well, the ICE is in play, installed, and unrezzed. Forged Activation orders specifies unrezzed ice.

I see no conflict there.

Yes, FAO can be used to force the corp to rez the ice (or trash it). And the corp wouldn't get the 7 credit discount because the runner hasn't passed all the ice on the server. But at the same time it also isn't trashed if you can afford the rez cost, though you can't force the runner to encounter it as it isn't triggered by Awakening Center.

That means that there is no point for the corp to rez a piece of ice hosted on awakening center if it is hit by FAO

I thought Forged activation order said "corp must rez or trash ice", and there is not Negative effect on Awakening Center, if could choose, corp must choose rez or trash it.

right?

btw, if corp could rez the ice and that ice still hosted on Awakening Center, how to resolve the effect of Awakening Center that whenever runner passes all of ice protecting this server....?

That's exactly what nungunz was trying to explain.

You can choose an ice on Awakening Center as the target for Forged Activation Orders.

The corp can trash the ice or rez it. If they rez it, it no longer works with the ability of Awakening Center, because that ability requires an unrezzed piece of ice. Therefore, the corporation never gains anything from rezzing the piece of ice on Awakening Center and will probably always choose trash it instead.