Isomorphic iff potentially conjugate

For just one pair of isomorphic subgroups
Suppose $$G$$ is a group and $$H, K \le G$$ are isomorphic groups, i.e., there is an isomorphism of groups, say $$\sigma$$, from $$H$$ to $$K$$ (Note that this isomorphism need not arise from an automorphism of $$G$$, so $$H$$ and $$K$$ need not be automorphic subgroups).

Then, there exists a group $$L$$ containing $$G$$ such that $$H, K$$ are conjugate subgroups inside $$L$$, and the induced isomorphism from $$H$$ to $$K$$ by that conjugating element equals $$\sigma$$.

For a collection of many pairs of isomorphisms between subgroups
Suppose $$G$$ is a group, $$I$$ is an indexing set, and $$H_i \cong K_i$$ are pairs of isomorphic subgroups of $$G$$ for each $$i \in I$$. Let $$\sigma_i: H_i \to K_i$$ be an isomorphism for each $$i \in I$$.

Then, there exists a group $$L$$ containing $$G$$ as a subgroup such that $$H_i$$ and $$K_i$$ are conjugate subgroups in $$L$$ for each $$i \in I$$. More specifically, we can find $$g_i, i \in I$$ such that the map induced by conjugation by $$g_i$$ induces the isomorphism $$\sigma_i$$.

Moreover, there is a natural construction of such a group $$L$$, called a HNN-extension. In the case that $$G$$ is an torsion-free group, we can ensure that the group $$L$$ is also torsion-free.

For finite groups

 * Isomorphic iff potentially conjugate in finite: This construction works when the original group is finite and yields a bigger group that is also finite.

Facts about automorphisms extending to inner automorphisms

 * Inner automorphism to automorphism is right tight for normality: In other words, if $$\sigma$$ is an automorphism of $$G$$, there exists a group $$L$$ containing $$G$$ as a normal subgroup, and an inner automorphism of $$L$$ whose restriction to $$G$$ equals $$\sigma$$.
 * Left transiter of normal is characteristic: A direct application of the fact that any automorphism of a group extends to an inner automorphism in a bigger group containing it as a normal subgroup. This says that $$H \le G$$ is such that (whenever $$G$$ is normal in $$L$$, $$H$$ is also normal in $$L$$) if and only if $$H$$ is characteristic in $$G$$.
 * Characteristic of normal implies normal

Facts about injective endomorphisms

 * Every injective endomorphism arises as the restriction of an inner automorphism

Applications

 * Same order iff potentially conjugate: $$x,y \in G$$ are such that $$x,y$$ have the same order if and only if then there is a group $$L$$ containing $$G$$ in which $$x$$ and $$y$$ are conjugate elements. This is a direct application based on looking at the cyclic subgroups $$\langle x \rangle$$ and $$\langle y \rangle$$.
 * Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes
 * Every torsion-free group is a subgroup of a simple torsion-free group