Can I activate an already exhausted group with this card?

No.

I was hoping for the same thing when I first saw it.

Basically, it's primary use is to pair it with something like a droid (like c3p0) to focus a figure on the turn you activate your other attacking hero or squad or maybe with someone like Gideon and something else to get in those extra dice when you think it's needed to finish a target off before they can activate whatever you're attacking before it can escape/heal/counterattack. So helpful when your target *must* go and you can risk activating two cards at that moment.

* Edit: Was speaking from my perspective having mostly played Rebels...other obvious use would clearly be two squads of regular Stormtroopers, all six attacking that round.