Complexity of the coset intersection problem
Moderately exponential time and $\mathsf{coAM}$ (for the opposite of the problem as stated: Coset Intersection is typically considered to have a "yes" answer if the cosets intersect, opposite of how it's stated in the OQ.)

Luks 1999 (free author's copy) gave a $2^{O(n)}$-time algorithm, while Babai (see his 1983 Ph.D. thesis, also Babai-Kantor-Luks FOCS 1983, and a to-appear journal version) gave a $2^{\tilde{O}(\sqrt{n})}$-time algorithm, which remains the best known to date. Since graph isomorphism reduces to quadratic-sized coset intersection, improving this to $2^{\tilde{O}(n^{1/4-\epsilon})}$ would improve the state of the art for graph isomorphism.