Inner implies 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., inner automorphism) must also satisfy the second automorphism property (i.e., extensible automorphism)
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Statement

Verbal statement

Any inner automorphism of a group is extensible.

Statement with symbols

Let $G\le H$ be groups and $\sigma$ an inner automorphism of $G$, there is an automorphism ${\sigma }^{\prime }$ of $H$ such that the restriction of ${\sigma }^{\prime }<math>to<math>G$ is $\sigma$.

Related facts

Converse

• Extensible implies inner
• Finite-extensible implies inner

Proof

The proof in fact follows from the result:

Inner is extensibility-stable

and the fact that every inner automorphism is an automorphism.