Every injective endomorphism arises as the restriction of an inner automorphism

Statement

Suppose $G$ is a group and $\sigma$ is an Injective endomorphism (?) of $G$. Then, there exists a group $K$ containing $G$ as a subgroup, as well as an Inner automorphism (?) $\alpha$ of $K$, such that the restriction of $\alpha$ to $G$ equals $\sigma$.

Notice that any endomorphism that arises as the restriction of an inner automorphism must be injective, hence this result is tight.

Related facts

• Inner automorphism to automorphism is right tight for normality: Every automorphism arises as the restriction of an inner automorphism from a bigger group in which the given group is normal.
• Left transiter of normal is characteristic
• Characteristic of normal implies normal
• Isomorphic iff potentially conjugate: Any isomorphism between two subgroups of a group can be realized as the restriction of an inner automorphism in some bigger group. This uses the natural construction of HNN-extensions.