Finite-extensible implies subgroup-conjugating

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., subgroup-conjugating automorphism)
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Statement

Suppose $G$ is a finite group and $\sigma$ is a finite-extensible automorphism of $G$. In other words, for any finite group $H$ containing $G$, there is an automorphism $\sigma '$ of $H$ whose restriction to $G$ equals $\sigma$.

Then, $\sigma$ is a subgroup-conjugating automorphism of $G$: it sends every subgroup of $G$ to a conjugate subgroup.

This is a partial result towards the finite-extensible automorphisms conjecture.

Related facts

• Extensible implies subgroup-conjugating: Essentially, the same proof idea works.
• Finite-extensible implies class-preserving