Finite-extensible implies subgroup-conjugating

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