Hall-extensible automorphism

This is a variation of extensible automorphism|Find other variations of extensible automorphism |

Definition

Let $G$ be a finite group and $\sigma$ be an automorphism of $G$. We say that $\sigma$ is a Hall-extensible automorphism of $G$ if, whenever $K$ is a finite group containing $G$ as a Hall subgroup, there exists an automorphism ${\sigma }^{\prime }$ of $K$ such that the restriction of ${\sigma }^{\prime }$ to $G$ equals $\sigma$.

Relation with other properties

Stronger properties

• Finite-extensible automorphism
• Inner automorphism (when working with a finite group).

Weaker properties

• Hall-quotient-pullbackable automorphism
• Class-preserving automorphism
• Normal automorphism