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$EG$ is also the universal free $G$-space, meaning that, if $X$ is a free $G$-space (let's assume of the $G$-homotopy type of a $G$-CW complex), there is, up to $G$-homotopy, a unique $G$-map $X\to EG$. Taking quotients, you get a map $X/G \to BG$ which induces the action of $H^\ast(BG)$ on $H^\ast(X)$ with any coefficients. Edited to add: I really should ...

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