A generalization of the Goresky-Klapper conjecture, Part I

For a fixed integer $n\geq 2,$ we show that a permutation of the least residues mod $p$ of the form $f(x)=Ax^k$ mod $p$ cannot map a residue class mod $n$ to just one residue class mod $n$ once $p$ is sufficiently large, other than the maps $f(x)=\pm x$ mod $p$ when $n$ is even and $f(x)=\pm x$ or $\pm x^{(p+1)/2}$ mod $p$ when $n$ is odd.