A generalization of the Goresky-Klapper conjecture, Part II

Suppose that $f(x)=A{x}^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $A{x}^{(p+1)\mathrm{/}2}$ mod $p$ we show that for fixed $n\ge 2$ the image of each residue class mod $n$ contains elements from every residue classe mod $n$, once $p$ is sufficiently large. If $f(x)=Ax$ mod $p$, then for each $p$ and $n$ there will be exactly $(1+o(1))\frac{6}{{\pi }^2}{n}^2$ readily describable values of $A$ for which the image of some residue class mod $n$ misses at least one residue class mod $n,$ even when $p$ is large relative to $n$. A similar situation holds for $f(x)=A{x}^{(p+1)\mathrm{/}2}$ mod $p$.