Factions don't intrinsically support any Power.
Instead, a Power influences a star system, and the faction officials in that star system are therefore influenced a little by that Power. But the same faction's officials in a different star system might find it necessary to follow the requests of another Power. Whether or not they actually like this deal or are just saying what they need to to survive is a different question.
In Powerplay 1, the controlling power gave various bonuses which would have been - at least on paper - useful to turn this "soft" influence into later putting an Imperial faction in charge of the stations. That never actually worked all that well in practice, especially not for Aisling Duval, so that's been dropped in Powerplay 2.
To an extent the superpowers in the ED setting - even ignoring Powerplay - aren't all that strong anyway. Federal factions will go to war with each other despite sharing a superpower, both the Federation and Empire have lost thousands of systems to independents in the last decade. The story is arguably one of significant decline and fall - and with the increased hostility in PP2, perhaps even the later breakup of the superpowers - and perhaps, in that sort of situation, a Federal official might find a reformist Imperial better able to provide their system with aid supplies than the distant (and until very recently military-focused) Federal Congress can. They're not going to officially defect - too messy, too likely to lead to open war - but if letting the Imperial Princess put a few posters up gets them the trade and aid deals they need, they can hardly afford not to. If Winters or Archer objects to that, they need to be making a better offer (and a few targeted assassinations of Duval's agents in the system wouldn't go amiss, either)