Difference sets disjoint from a subgroup II: groups of order $4p^2$

Abstract

We study finite groups $G$ having a normal subgroup $H$ and $D \subset G \setminus H, D \cap D^{-1}=\emptyset,$ such that the multiset $\{ xy^{-1}:x,y \in D\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\it DRAD difference set}). We show that there are no such groups of order $4p^2$, where $p$ is an odd prime.