Does the automaton's energy beam do double damage on a critical hit? Or just the persistent damage?
I think you mistakenly made the thread twice, so maybe you should erase the eye beam one?
Anyway to answer the question: Yes, Energy Beams do double damage on a critical hit, because that is an effect of the Strike action, not the weapon itself. The persistent fire damage is...I guess the equivalent of the critical specialisation effect, since the Energy Beams have no weapon group and specifically say they can't apply critical specialisation.
Agreed. The Energy Beam is an unarmed attack. So you make a Strike with it as normal for unarmed attacks, and it gets the benefits of a crit with a Strike.
You wouldn't double the persistent damage on a crit because you don't double due to a crit any damage that only happens in the first place because of the crit.