  We investigate pairs of permutations $F,G$ of $\mathbb{F}_{p^n}$ such that
$F(x+a)-G(x)$ is a permutation for every $a\in\mathbb{F}_{p^n}$. We show that
necessarily $G(x) = \wp(F(x))$ for some complete mapping $-\wp$ of
$\mathbb{F}_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear
(P$\wp$N) function. If $\wp(x) = cx$, then $F$ is a PcN function, which have
been considered in the literature, lately. With a binary operation on
$\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain a
quasigroup, and show that the graph of a P$\wp$N function $F$ is a difference
set in the respective quasigroup. We further point to variants of symmetric
designs obtained from such quasigroup difference sets. Finally, we analyze an
equivalence (naturally defined via the automorphism group of the respective
quasigroup) for P$\wp$N functions, respectively, the difference sets in the
corresponding quasigroup.
