Inner is extensibility-stable

Statement

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 ${\sigma }^{\prime }$ of $H$ such that the restriction of ${\sigma }^{\prime }$ to $G$ is $\sigma$.

Proof

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 $\sigma (x)=gx{g}^{-1}$.

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

This proves the result.