Model theory of fields with virtually free group actions

For a group $G$, we define the notion of a $G$-kernel and show that the properties of $G$-kernels are closely related with the existence of a model companion of the theory of Galois actions of $G$. Using Bass-Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP$_2$.