Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.

Is it always true that the number of ordered pairs $(g, h)$ of elements of $G$ satisfying $E$ is a multiple of $|G|$?

I know that it is true in many cases and I can't find any counterexamples. Does anybody know a simple counterexample, or a suitable reference for this question?