Powerfully embedded subgroups of extensions of powerful pro-$p$ groups and its applications to automorphisms of finite groups
One of the aims of this paper is to obtain structural results showing that powerful subgroups are abundant in pro-$p$ groups admitting certain powerful quotients. In particular, we obtain an analogue of Baer's theorem for powerful pro-$p$ groups, namely that the powerfulness of $H/Z_{n-1}(H)$ implies that the $n$th terms of both the lower $p$-series and the lower central series of $H$ are powerfully embedded in $H$. As a consequence, we obtain that if $H$ is a finitely generated pro-$p$ group and $H/Z_n(H)$ is a $p$-adic analytic pro-$p$ group for some positive integer $n$, then $H$ is a $p$-adic analytic pro-$p$ group. We also study crossed squares of powerful $p$-groups, establishing that if $μ: M \to G$ is a crossed module with $M$ a finite powerful $p$-group and $G$ a finite $p$-group, and if $μ(M)$ is powerfully embedded in $G$, then both $M \otimes G$ and $M \otimes^{p} G$ are powerful. Generalizing the commutator and the Frattini subgroups via the action of automorphisms, we study relative autocommutator and the generalized Frattini subgroups $L_A(G)$ and $Φ_A(G)$, respectively, for subgroups $A$ with $\operatorname{Inn}(G) \leq A \leq \operatorname{Aut}(G)$. When $G$ is a finite powerful $p$-group and $A$ is chosen from $\operatorname{Aut}_Φ(G)$, $\operatorname{Aut}_c(G)$, or $\operatorname{IA}(G)$, we show that $\operatorname{Inn}(G)$ is powerfully embedded in $A$. As a consequence, both $L_A(G)$ and $Φ_A(G)$ are powerful subgroups of $G$.