  A {\em Wakeford pairing} from $S$ onto $T$ is a bijection $\phi : S \to T$
such that $x\phi(x)\notin T,$ for every $x\in S.$ The number of such pairings
will be denoted by $\mu(S,T)$.
  Let $A$ and $ B$ be finite subsets of a group $G$ with $1\notin B$ and
$|A|=|B|.$ Also assume that the order of every element of $B$ is $\ge |B|$.
Extending results due to Losonczy and Eliahou-Lecouvey, we show that
$\mu(B,A)\neq 0.$ Moreover we show that $\mu(B,A)\ge \min
\{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ unless there is $a\in A$
such that $|Aa^{-1}\cap B|=|B|-1$ or $Aa^{-1}$ is a progression. In particular,
either $\mu(B,B) \ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$
or for some $a\in B,$ $Ba^{-1}$ is a progression.
