  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 ${\mathbb Z}_n$.
