Let $M$ be a smooth manifold and $G$ a finite group of automorphisms acting properly on it. If $\pi:M\to M/G$ is the projection map, prove that $\pi^*:H_{dR}^k(M/G)\to H_{dR}^k(M)$ is injective.

I know how to prove that $\pi^*:\Omega^k (M/G)\to\Omega^k (M) $ is injective using the fact that $\pi$ is a surjective submersion. But how do I prove injectiveness in cohomology?