Classify all groups containing isomorphic copy of $\mathbb{Z}$ of index $2$.

Hint. Let $G$ be a group in which $\mathbb{Z}$ is a subgroup of index $2$.

$\hspace{10pt}\text{S}{\small \text{TEP}}\text{ I}.\hspace{8pt}$ Prove that subgroups of index $2$ are necessarily normal.

$\hspace{10pt}\text{S}{\small \text{TEP}}\text{ II}.\hspace{5pt}$ This means that $G$ acts by automorphisms on $\mathbb{Z}$. What are the automorphisms of $\mathbb{Z}$?

$\hspace{10pt}\text{S}{\small \text{TEP}}\text{ III}.\hspace{2pt}$ If $G=\mathbb{Z}K$, what are the options for $K$? Relate this to the different options in $\text{II}$.

Note. Be careful in $\text{S}{\small \text{TEP}}\text{ III}$ to consider the case where the extension is nonsplit (that is, $G$ is not a semidirect product of the form $\mathbb{Z}\rtimes K$). If you get stuck here, try constructing group presentations.

If your group is torsion-free, then it is isomorphic to $\mathbb{Z}$ (for a proof, see here).

If it has torsion, then it has an element of order $2$ (can you see why?). This element generates a subgroup $H$ of order $2$, and your group is the semidirect product $\mathbb{Z}\rtimes H$. This gives either a trivial action ($\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$), or a non-trivial action (infinite dihedral group).