If $G$ is a torsion-free group, such that there exists a subgroup $H \cong \mathbb Z$ of finite index, then $G \cong$?

Question

If $G$ is a torsion-free group, such that there exists a subgroup $H \cong \mathbb Z$ of finite index, then $G \cong$?

I guess $G\cong \mathbb Z$. It clearly satisfies the requirements, but I'm afraid that there are multiple possibilities. Is this a characterization of $\mathbb Z$? Thanks in advance.