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 $H/Z_{n-1}(H)$ implie