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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 pairingswill 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.

On Group bijections $\phi $ with $\phi(B)=A$ and $\forall a\in B, a\phi(a) \notin A$