Residual properties of automorphisms groups and split extensions

Let a group $G$ satisfy condition A: for every positive integer $n$ the number of all subgroups of the group $G$ of index $n$ is finite. We prove that if $G$ is virtually residually finite $p$-group for some prime $p$, then the automorphism group of the group $G$ is virtually residually finite $p$-group. A similar result is obtained for a split extension of the group $G$ by virtually residually finite $p$-group. Moreover, we prove that if the group $G$ is a virtually residually finite nilpotent $\pi$-group for some finite set $\pi$ of primes, then the automorphism group of the group $G$ and the split extension of the group $G$ by a virtually residually finite nilpotent $\pi$-group are virtually residually finite nilpotent $\pi$-groups.