Hall-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 defining ingredient::Hall subgroup, there exists an automorphism $$\sigma'$$ of $$K$$ such that the restriction of $$\sigma'$$ to $$G$$ equals $$\sigma$$.

Stronger properties

 * Weaker than::Finite-extensible automorphism
 * Weaker than::Inner automorphism (when working with a finite group).

Weaker properties

 * Stronger than::Hall-quotient-pullbackable automorphism
 * Stronger than::Class-preserving automorphism
 * Stronger than::Normal automorphism