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To elaborate a little on Kostya's answer, for the case where $G$ acts freely you may want to look up the Cartan-Leray spectral sequence (eg Chapter 7 of the book by Brown). Also, transfer homomorphisms are a useful elementary tool (see section 3.G of Hatcher's book).
When $G$ doesn't act freely you're into the realm of orbifolds, about which I have much to learn!