Let $G$ be an infinite group with identity element $e$. Consider the first order theory of faithful $G$-sets (sets along with a faithful $G$ action) $(X,.)$ which along with the axioms of set theory, has the following set of axioms : For every $g\in G$, consider the unary function symbol $g.$ and collect the sentences $\forall x (g.(h.x)=(gh).x)$; $\forall x (e.x=x)$, and for every non-identity element $g$ in $G$, collect the sentences $\exists x (g.x\ne x)$.
Is this first order theory of faithful $G$-sets complete ? Is it model complete ?