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., extensible automorphism) must also satisfy the second automorphism property (i.e., subgroup-conjugating automorphism)
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Statement

Every extensible automorphism of a group must be a subgroup-conjugating automorphism: it must sned every subgroup to a conjugate subgroup.

Related facts

Corollaries

Facts used

  1. Extensible implies permutation-extensible
  2. Equivalence of definitions of subgroup-conjugating automorphism: This essentially shows that the notion of permutation-extensible is equivalent to the notion of subgroup-conjugating.

Proof

The proof follows by piecing together facts (1) and (2).