I know the proposition that says:
Let $\lambda$ be a homomorphism from a finite group $G$ into $\mathbb{C}^{\times}$. Suppose that $G$ acts on some finite set $\Omega$ and let $M$ be the number of $\lambda$-good orbits for this action. Then $M=\frac{1}{|G|}\sum_{g\in G}\lambda(g)\pi(g),$ where $\pi$ is the permutation character associated with the action.
Using this, I need to prove:
Let $G$ be a finite group acting on a finite set $\Omega$, and let $\lambda$ be an arbitrary homomorphism from $G$ into $\mathbb{C}^{\times}$. Then for each positive integer $a$: $\sum_{g\in G}\lambda(g)a^{c(g)} \equiv 0 \pmod{|G|},$ where $c(g)$ is the number of orbits of $\langle g \rangle$ on $\Omega$.