Find all groups $G$ with a normal subgroup $H\simeq\Bbb{Z}$ such that $[G:H]=2$

Find all groups $G$ with a normal subgroup $H\simeq\Bbb{Z}$ and such that $[G:H]=2$.

How do I classify such groups? One obvious example is $\Bbb{Z}/2\Bbb{Z}$?

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Identical question math.stackexchange.com/questions/590651/… – Tobias Kildetoft Dec 3 '13 at 9:17

Let $h$ be a generator $\mathbb Z\cong H \subset G$, and let $g\in G$ with $g\not\in H$. Then $g$ and $h$ generate $G$. Moreover, $g^2=h^a$ for some $a\in \mathbb N$ because $G/H \cong \mathbb Z /2 \mathbb Z$, and $ghg^{-1}=h^b$ for some $b\in \mathbb N$ because $H$ is normal. Therefore, the groups you are looking for are classified in some sense by the ordered pairs $(a,b)$. However, this is incomplete because