Do subgroups have "two sided bases"?

If $H$ is a subgroup of finite index in a group $G$, there is a subset $\mathcal B$ of $G$ which serves both as a set of representatives for the left cosets of $H$ in $G$ and as a set of representatives for the right cosets of $H$ in $G$. (See, for example, Theorem 3, §4, Chap. I, in the book The Theory of groups by H. Zassenhaus) That should do it, no?

It was recently proved that every regular subfactor (resp., depth 2 subfactor with simple relative commutant) admits a two-sided basis - see http://nyjm.albany.edu/j/2020/26-37.html (resp., https://arxiv.org/abs/2102.01462).