On the Cohen–Lusk theorem

Let $G$ be a finite group and $X$ be a $G$-space. For a map $f\colon X\to\mathbb R^m$, the partial coincidence set $A(f,k)$, $k\leq|G|$, is the set of points $x\in X$ such that there exist $k$ elements $g_1,\dots,g_k$ of the group $G$, for which $f(g_1x)=\dots=f(g_kx)$ hold. We prove that the partial coincidence set is nonempty for $G=\mathbb Z_p^n$ under some additional assumptions.