Consider a finite group $G$ and subgroups $\{H_i\}_{i=0,\ldots,n-1}$ (defined up to conjugacy) such that $G$ acts transitively on each $G/H_i$ [edit: in the canonical way]. Consider the induced action of $G$ on the cartesian product $\prod_{0\leq i<n} G/H_i$. This action is not necessarily transitive.
Question: Is there a nice description of the orbits of this action and their (conjugacy classes of) stabilizers?
If I remember correctly, for the cartesian product of two transitive actions, the orbits are in one-to-one correspondence with double cosets $H_0\backslash G/H_1$. Not sure how to get the stabilizers though.