Powerfully embedded subgroups of extensions of powerful pro-$p$ 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 $H\mathrm{/}{Z}_{n-1}(H)$ 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 $H\mathrm{/}{Z}_n(H)$ 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 $\mu \text{\hspace{0.17em}⁣}:M\to G$ is a crossed module with $M$ a finite powerful $p$-group and $G$ a finite $p$-group, and if $\mu (M)$ is powerfully embedded in $G$, then both $M\otimes G$ and $M{\otimes }^{p}G$ are powerful.