Relative difference sets fixed by inversion (III)—Cocycle theoretical approach

In this paper, we give a characterization of a group G which contains a semiregular relative difference set R relative to a central subgroup N   containing the commutator subgroup [G,G][G,G] of G   such that 1∈R1∈R and rRr=RrRr=R for all r∈Rr∈R. In particular, these relative difference sets are fixed by inversion and inner automorphisms of the group are multipliers. We also present a construction of such relative difference sets.