Not sure if you're already aware of the exact sequence which implies Hurewicz, but we have
$$H_{i+1}(X)\to H_{i+1}(\pi)\to (H_i(X))_\pi\to H_i(X)\to H_i(\pi)\to0$$ where $H_*(\pi)$ is discrete group cohomology and $\pi$ acts on $H_i(X)$ via the universal cover over $X$ (I mean that's one way to phrase the action). See classic book of Spanier, or can be proved by spectral sequence, c.f. Ken Brown's bible "Cohomology of Groups" (exercise IV.5.1 and exercise VII.7.6).