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I learned that "If $G$ is a finite group acting freely and continuously on $S^n$, the sphere then $G$ has periodic cohomology".

My question is: Are there any other similar theorems relating the free action of a finite group and its cohomology?

maybe this question is too vague.

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You may want to check out Alex Adem's paper:

Adem, Alejandro, Cohomological restrictions on finite group actions, J. Pure Appl. Algebra 54, No.2-3, 117-139 (1988). ZBL0686.57023. If you look at the papers which cite this one on Google Scholar some wisdom may likewise ensue.

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