Here is another interesting question that I can't answer on my own.
Let $G$ be a countable, discrete group such that for any subgroup $H$ of $G$ and any element $s$ of $G$ we have $[H : sHt]$ is finite and $[H : tHs]$ is finite where $st=ts=1$, i.e. $t$ is the inverse of $s$. Then $G$ is amenable.
Note that in the case when the indexes above are exactly equal $1$ then $G$ is Hamiltonian which is solvable and therefore amenable.
Thanks in advance.