Model theoretic dynamics in Galois fashion

We investigate existentially closed models (of a quite arbitrary theory) equipped which an action of a fixed group G. We embed these structures in a monster model D of some well-rounded theory and describe them as PAC substructures of D. Assuming that the class of these existentially closed models is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas - similar to ACFA. Moreover, it codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imaginaries.