Inner is extensibility-stable

Verbal statement
Any inner automorphism of a subgroup lifts to an inner automorphism of the whole group.

Symbolic statement
Let $$G \le H$$ be groups and $$\sigma$$ be an inner automorphism of $$G$$. Then, there exists an inner automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

Hands-on proof
We are given $$G \le H$$ and an inner automorphism $$\sigma$$ of $$G$$. Since $$\sigma$$ is an inner automorphism of $$G$$, there exists $$g \in G$$ such that $$\sigma(x) = gxg^{-1}$$.

Now consider the inner automorphism of $$H$$ defined via conjugation by $$g$$, that is, the map $$\sigma' = x \mapsto gxg^{-1}$$ over $$H$$. Clearly, this is an inner automorphism of $$H$$, and its restriction to $$G$$ is the same map $$\sigma$$.

This proves the result.