I am looking for examples of the following situation:

$G$ is an infinite group. Every two finite index proper subgroups of $G$ are isomorphic.

The only examples that I have now are (1) $\mathbb{Z}^n$ for $n$ any cardinal number, (2) groups that have at most one proper infinite subgroup of finite index (thanks to Sasha Anan'in's and Derek Holt's comments), (3) direct product of groups in (1) and (2).

Are these the only examples for the above situation?

Is there a classification for groups with this property?

Can we find examples with various properties? For example, $G$ is not amenable, etc.

Any related reference will be appreciated.