  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.
