  Motivated by a connection between semi-regular relative difference sets and
mutually unbiased bases, we study relative difference sets with parameters
$(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We
prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any
group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can
only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we
construct a family of non-abelian relative difference sets with parameters
$(4q,q,4q,4)$, where $q$ is an odd prime power greater than 9 and $q\equiv 1$
(mod 4). When $q=p$ is a prime, $p>9$, and $p\equiv$ 1 (mod 4), the
$(4p,p,4p,4)$ non-abelian relative difference sets constructed here are
genuinely non-abelian in the sense that there does not exist an abelian
relative difference set with the same parameters.
