Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible

Statement

Suppose $P$ is a finite $p$-group such that the center $Z(P)$ is a cyclic group of order $p$. Suppose, further, that the inner automorphism group $\operatorname{Inn}(P)$ is a maximal subgroup in the normal closure of any $p$-Sylow subgroup in the automorphism group $\operatorname{Aut}(P)$. Then, if $\sigma$ is a $p$-automorphism of $P$ (i.e., an automorphism whose order is a power of $p$), and $Q$ is a group containing $P$ as a normal subgroup, $\sigma$ can be extended to an automorphism $\sigma'$ of $Q$.

Examples

The prototypical example of this is where $P$ is dihedral group:D8. In this case, $\operatorname{Aut}(P)$ is also a dihedral group of order eight, with the inner automorphism group a Klein four-subgroup. Clearly, all the conditions for the statement are satisfied, and hence, all the $2$-automorphisms of the dihedral group (which in this case means all automorphisms) can be extended to any group containing this as a normal subgroup.

Related facts

• Centerless and maximal in automorphism group implies every automorphism is normal-extensible
• Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible