Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).
Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?
Remark: It is true for $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see this paper Corollary 4.33). It's also true (using the function is_cohen_macaulay on SAGE) for the three first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here (my desktop isn't enough powerful for checking the fourth one).
