Let $G$ be the cyclic group of order $2$ acting by inversion on $\mathbb{Z}$. Show $|H^1(G,\mathbb{Z})|=2$. A hint is provided: if $E=\mathbb{Z} \rtimes G$ then every element in $E - \mathbb{Z}$ has order $2$ and there are two conjugacy classes.
To solve this its clearly enough to classify the splittings of the exact sequence $\mathbb{Z} \to E \to G$ up to $G$-conjugacy. I've shown the first part of the hint but I'm not sure how to do the second part. Also, with that, how do I get the splittings? I feel like I'm missing something obvious.