Here's a problem I had last semester in my grad Algebra class which really helped me understand short exact sequences that split.

Let $\displaystyle D_{\infty}$ be a group with presentation $\displaystyle <a,b\mid a^2=b^2=e> $

Show that $\displaystyle G \simeq\mathbb{Z}\rtimes\mathbb{Z}/2$.

Hint: Find a short exact sequence which splits. The sequence should be clear from the isomorphism.