Some smaller examples: sometimes a group is best understood by embedding it in a larger group. One can use the embedding of a finite group in a large enough $GL(n, \mathbb{C})$ as a very general example (representation theory is quite powerful, even just in characteristic 0), but some far more localized examples:

The reason plenty of subgroups of $S_{6}$ have two non-conjugate embeddings in $S_{6}$ is explainable by embedding $S_{6}$ in $Aut(S_{6})$. Likewise for $AGL_{3}(2)$ and $M_{12}$.

Much of the structure of $M_{22}$ and $M_{23}$ is most easily understood in terms of $M_{24}$.