Model theoretic dynamics in Galois fashion

We fix a monster model $\mathfrak{D}$ of some stable theory and investigate substructures of $\mathfrak{D}$ which are existentially closed within the class of substructures equipped with an action of a fixed group G. We describe them as PAC substructures of $\mathfrak{D}$ and obtain results related to Galois theory.Assuming that the class of these existentially closed substructures is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas. Moreover, this theory codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imaginaries.