On the virtual potency of automorphism groups and split extensions

We obtain some sufficient conditions for potency and virtual potency for automorphism groups and the split extensions of some groups. In particular, considering a finitely generated group $G$ residually $p$-finite for every prime $p$, we prove that each split extension of $G$ by a torsion-free potent group is a potent group, and if the abelianization rank of $G$ is at most $2$ then the automorphism group of $G$ is virtually potent. As a corollary, we derive the necessary and sufficient conditions of virtual potency for certain generalized free products and HNN-extensions.