In brief: suppose that the left and right cosets coincides as two coverings of the group as a set, does it imply that the subgroup is normal?
Detailed: Let $G$ and $H$ be a finite group and its subgroup, respectively. If $H$ is normal in $G$ then the left cosets coincides with the right ones, i.e. $gH=Hg$ for any $g\in G$.
Suppose that for any $a\in G$ there is $b \in G$ such that $aH=Hb$, i.e. left and right cosets 'coincide as partitions of a group into subsets', but it is not required that $aH=Ha$. I wonder if one can derive that $H$ is actually a normal subgroup or give a counter example?
