We introduce two refinements of the class of $5/2$-groups, inspired by the classes of automorphism groups of configurations and automorphism groups of unit circulant digraphs. We show that both of these classes have the property that any two regular cyclic subgroups of a group $G$ in either of these classes are conjugate in $G$. This generalizes two results in the literature (and simplifies their proofs) that show that symmetric configurations and unit circulant digraphs are isomorphic if and only if they are isomorphic by a group automorphism of ${\mathbb Z}_n$.