I would like to show that if the sequence:

$\displaystyle 0 \to \mathbb{Z} \to G \to \mathbb{Z}/2\mathbb{Z} \to 0$

is exact, then G need not necessarily be isomorphic to $\displaystyle (\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z}$. Just one example of G that is not isomorphic to this (where the above sequence is still exact) would be enough. I've tried many and I can't seem to find a single valid one.. any hints?