Hall-semidirectly extensible automorphism

Definition
Let $$G$$ be a finite group and $$\sigma$$ be an automorphism of $$G$$. $$\sigma$$ is termed Hall-semidirectly extensible if for every group $$K$$ containing G as a defining ingredient::Hall retract, i.e., containing $$G$$ as a Hall subgroup with a normal complement, there exists an automorphism $$\sigma'$$ of $$K$$ whose restriction to $$G$$ is $$\sigma$$.

Stronger properties

 * Weaker than::Finite-characteristic-semidirectly extensible automorphism
 * Weaker than::Hall-extensible automorphism

Weaker properties

 * Stronger than::Class-preserving automorphism