Finite-extensible implies Hall-semidirectly extensible

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., finite-extensible automorphism) must also satisfy the second automorphism property (i.e., Hall-semidirectly extensible automorphism)
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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 $G$.

Definitions used

Finite-extensible automorphism

Further information: Finite-extensible automorphism

Hall-semidirectly extensible automorphism

Further information: Hall-semidirectly extensible automorphism

Related facts

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.