Inner implies extensible

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 $$\sigma'$$ of $$H$$ such that the restriction of \sigma' to $$G$$ is $$\sigma$$.

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.