Let $G$ be a finite group acting on a manifold $M$ without fixed point. The standard Leray-Cartan-Serre spectral sequence argument shows that $ H^k(M,\mathbb{Q})^G\cong H^k(M/G,\mathbb{Q}). $ This in particular means that rank $H^k(M,\mathbb{Z})^G$=rank $H^k(M/G,\mathbb{Z})$.
We also have a natural map $\pi^{*}:H^k(M/G,\mathbb{Z})\rightarrow H^k(M,\mathbb{Z})^G$ via the quotient map $\pi:M\rightarrow M/G$.
Is it true that $\pi^{*}$ is injection (and hence that $H^k(M/G,\mathbb{Z})\subset H^k(M,\mathbb{Z})^G$ is of finite index)?