P-normal-extensible automorphism

Definition

Suppose $p$ is a prime number and $P$ is a p-group (i.e., a group where the order of every element is a power of $p$). An automorphism $\sigma$ of $P$ is termed a $p$-normal-extensible automorphism if, for any $p$-group $Q$ containing $P$ as a normal subgroup, there exists an automorphism ${\sigma }^{\prime }$ of $Q$ whose restriction to $P$ equals $\sigma$.

Facts

• 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: Thus, for instance, all automorphisms of dihedral group:D8 are p-normal-extensible.