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

Galois actions of finitely generated groups rarely have model companions