If $G$ is a group and $\sigma$ is an injective endomorphism of $G$, there is a group $K$ containing $G$ such that $\sigma$ extends to an inner automorphism of $K$.

Even more generally, if two subgroups of a group are isomorphic, the group can be embedded in a bigger group where those two subgroups are conjugate via a conjugation map that extends the isomorphism. The construction used is called a HNN-extension, and it basically adjoins elements to the group that act by conjugation as the isomorphism. (This generalizes even further: given any number of isomorphisms between pairs of subgroups of a group, there is a group containing the group such that all these isomorphisms become conjugations in that bigger group.)

Thus, to find examples that answer your question, it is enough to find examples of a group that is isomorphic to a proper subgroup. For instance, if we consider the example of the group of integers isomorphic to the subgroup of even integers, the corresponding HNN-extension is the Baumslag–Solitar group mentioned above. 

Incidentally, the statement above (that any two isomorphic subgroups of a group become conjugate in some bigger group) is also true when we restrict to finite groups, though this does not give any examples for the question you are interested in because no finite subgroup can be isomorphic to a proper subgroup.

See <a href="http://groupprops.subwiki.org/wiki/Isomorphic_iff_potentially_conjugate">isomorphic iff potentially conjugate</a> and <a href="https://groupprops.subwiki.org/wiki/Isomorphic_iff_potentially_conjugate_in_finite">isomorphic iff potentially conjugate in finite</a> on the Groupprops wiki for more notes on these.