Every injective endomorphism arises as the restriction of an inner automorphism

Statement
Suppose $$G$$ is a group and $$\sigma$$ is an fact about::injective endomorphism of $$G$$. Then, there exists a group $$K$$ containing $$G$$ as a subgroup, as well as an fact about::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.