To consider some examples, if *G* = *H* × *K*, then *G* is an extension of both *H* and *K*. More generally, if *G* is a semidirect product of *K* and *H*, then *G* is an extension of *K* by *H*, so such products as the wreath product provide further examples of extensions.

The question of what groups *G* are extensions of *K* is called the **extension problem**, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {*A*_{i}}, where each *A*_{i+1} is an extension of *A*_{i} by some simple group. The classification of finite simple groups gives us a complete list of finite simple groups; so the solution to the extension problem gives us enough information to construct and classify all finite groups in general.

We can use the language of diagrams to provide a more flexible definition of extension: a group *G* is an extension of a group *K* by a group *H* if and only if there is an exact sequence:

- 1 →
*K*→*G*→*H*→ 1

*mention Ext functor; mention that extensions are known*