I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question.

Let $G$ be a finite group acting on a manifold $M$ without fixed point. The standard spectral sequence argument shows that $$ H^k(M,\mathbb{Q})^G\cong H^k(M/G,\mathbb{Q}). $$ This in particular shows 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 mod torsion (and hence that $H^k(M/G,\mathbb{Z})\subset H^k(M,\mathbb{Z})^G$ (mod torsions) is of finite index)? Or are there any relation between $\mathbb{Z}$-coefficient (co)homology groups?