Finite-extensible implies Hall-semidirectly extensible

Statement
Suppose $$G$$ is a finite group and $$\sigma$$ is a finite-extensible automorphism of $$G$$. Then, $$\sigma$$ is a Hall-semidirectly extensible automorphism of $$G$$.

Intermediate properties

 * Finite-semidirectly extensible automorphism is an automorphism that can always be extended from the retract part of a semidirect product.
 * Finite-characteristic-semidirectly extensible automorphism is an automorphism that can always be extended from the retract part of a semidirect product, as long as the normal complement is a characteristic subgroup.

Proof
This is direct from the definitions.