We study finite groups $G$ having a nontrivial subgroup $H$ and $D \subset G \setminus H$ such that (i) the multiset $\{ xy^{-1}:x,y \in D\}$ has every element that is not in $H$ occur the same number of times (such a $D$ is called a {\it relative difference set}); (ii) $G=D\cup D^{(-1)} \cup H$; (iii) $D \cap D^{(-1)} =\emptyset$. We show that $|H|=2$, that $H$ has to be normal, and that $G$ is a group with a single involution. We also show that $G$ cannot be abelian. We give examples of such groups, including certain dicyclic groups, by using results of Schmidt and Ito. We describe an infinite family of dicyclic groups with these relative difference sets, and classify which groups of order up to $72$ contain them. We also define a relative difference set in dicyclic groups having additional symmetries, and completely classify when these exist in generalized quaternion groups. We make connections to Schur rings and prove additional results.

Strong Gelfand pairs describe a relation between a group and a subgroup, using a relation between inner products of their characters. We find all strong Gelfand pairs of the dihedral and dicyclic groups, and several of the sporadic groups. We provide some results for the strong Gelfand pairs of the affine linear groups, in addition to the exceptional classical groups $\mathrm{Sp}_4(q)$ for $q$ a power of $2$.