Finite-extensible implies subgroup-conjugating

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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.

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