Exercise 28, § 5

Let $G$ be a group, $A$ and $B$ two subgroups of $G$ and $\phi$ an isomorphism of $A$ onto $B$. Show that there exists a group $G_1$ containing $G$ such that $\phi$ is the restriction of an inner automorphism of $G_1$ (use the permutation group of the set $G$). Show that, if $G$ is finite, $G_1$ can be chosen to be finite.