This is joint work with Özlem Beyarslan. For a fixed finitely generated group $G$, we consider actions of $G$ by field automorphisms. If the theory of such generic actions is first-order axiomatizable, then we say that $G$-TCF exists. It is well-known that $G$-TCF exists if $G$ is a free group (the theory ACFA$_n$), and it is also known that $G$-TCF exists for a finite $G$. On the other hand, it is also known that $(Z\times Z)$-TCF does not exist. Using Bass-Serre theory, we prove that if $G$ is virtually free, then $G$-TCF exists. Using the results of Chatzidakis and Hoffmann, we can also say for which G the obtained theory $G$-TCF is simple.