We study the $G-$ invariant version of perfectly meager sets (a generalization of the notion of AFC' sets). We find the necessary and sufficient conditions for the inclusion $AFC'_G \subseteq \mathcal{I}$. In particular, we partially characterize for which groups $G$ of automorphisms of the Cantor space every $\mathrm{AFC}'_{G}$ set is Lebesgue null.