Galois actions of finitely generated groups rarely have model companions

We show that if $G$ is a finitely generated group such that its profinite completion $\widehat{G}$ is far from being projective'' (that is the kernel of the universal Frattini cover of $\widehat{G}$ is not a small profinite group), then the class of existentially closed $G$-actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is far from being projective’‘, the main result of this paper corrects an error in our paper ``Model theory of fields with virtually free group actions’', Proc. London Math. Soc., (2) 118 (2019), 221–256 by showing the negation of Theorem 3.26 in that paper.