  Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod
$p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we
show that for fixed $n\geq 2$ 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
$f(x)=Ax^{(p+1)/2}$ mod $p$.
