It is of interest to me to find if there is a way to reconstruct the group knowing the (multi)-set of its conjugacy class sizes or, in other words, given the indices of all centralizers is there a unique group satisfying them (up to isomorphism)?
A piece of motivation was the similar statement for graphs involving degree sequences. It is known that non-isomorphic graphs with equal degree sequences do exist in that case.
The answer to my question in general is no as well since I've noticed that both dihedral group $D_8$ and Quaternion group $Q_8$ have the same multiset $\{ 1, 1, 2, 2, 2 \}$ although they are obviously not isomorphic.
However are there any well-known results in this area? E.g. for groups of particular order or with some kind of restriction applied?