  For nearly a century, mathematicians have been developing techniques for
constructing abelian automorphism groups of combinatorial objects, and,
conversely, constructing combinatorial objects from abelian groups. While
abelian groups are a natural place to start, recent computational evidence
strongly indicates that the vast majority of transitive automorphism groups of
combinatorial objects are nonabelian. This observation is the guiding
motivation for this paper. We propose a new method for constructing nonabelian
automorphism groups of combinatorial objects, which could be called the
\textit{combinatorial transfer method}, and we demonstrate its power by finding
(1) the first infinite families of nonabelian Denniston partial difference sets
(including nonabelian Denniston PDSs of odd order), (2) the first infinite
family of Spence difference sets in groups with a Sylow 3-subgroup that is
non-normal and not elementary abelian, (3) the first infinite families of
McFarland difference sets in groups with a Sylow $p$-subgroup that is
non-normal and is not elementary abelian, (4) new infinite families of partial
difference sets in nonabelian $p$-groups with large exponent, (5) an infinite
family of semiregular relative difference sets whose forbidden subgroup is
nonabelian, and (6) a converse to Dillon's Dihedral Trick in the PDS setting.
We hope this paper will lead to more techniques to explore this largely
unexplored topic.
