An automorphism fixing the centre pointwise induces a pairing-preserving automorphism of the central quotient, with kernel the inner automorphisms

Given: An extraspecial $p$-group $P$ with $Z(P)=⟨z⟩$, its commutator pairing $b_z$, and an automorphism $\alpha$ fixing $Z(P)$ pointwise.

For an extraspecial $p$-group $P$ with $Z(P)=⟨z⟩$, the commutator pairing is the map $b_z(\bar{x},\bar{y})\in \mathbb{Z}\mathrm{/}p$ determined by $[x,y]=z^{b_z(\bar{x},\bar{y})}$ (The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre).

For a finite $p$-group $P$ the following are equivalent: $P$ is extraspecial; $P$ is nonabelian, $\mathrm{\mid }Z(P)\mathrm{\mid }=p$ and $P\mathrm{/}Z(P)$ is elementary abelian; $P$ is nonabelian and $Z(P)=P^{\mathrm{\prime }}=\mathrm{\Phi }(P)$ has order $p$ (Three equivalent descriptions of an extraspecial $p$-group).

Every extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$ (An extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$).

For an extraspecial $p$-group of order $p^{1+2n}$, an automorphism acting trivially on its Frattini quotient is inner (An automorphism of an extraspecial $p$-group acting trivially on its Frattini quotient is inner).

Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$. (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$).

Inner automorphisms and $\mathrm{Inn}(G)$. (Inner automorphisms and $\mathrm{Inn}(G)$).

$[g,h]:=gh{g}^{-1}{h}^{-1}\mathrm{.}$ (Commutators $[g,h]=gh{g}^{-1}{h}^{-1}$ and the commutator subgroup $[G,G]$).