For *CW-complexes*, this is spelled out explicitly in *Ken Brown's bible* (as always): Proposition II.4.1 (which is the combination of Proposition I.4.2 and Proposition II.2.4), after using the algebraic/homological-definition of group cohomology in Section II.3 (effectively we have maps between their explicit $\mathbb{Z}G$-free resolutions of $\mathbb{Z}$ which you can then twist with local coefficients as clarified in Chapter III.1). For *general spaces*, this is Exercise II.4.2, which he gives hints and furthermore gives the foundational reference that writes it out (Eilenberg--Maclane's 1945 *"Relations between homology and homotopy
groups of spaces"*).

More generally, for any path-connected topological space X we can construct a discrete group G and K(G,1) inducing (co)homology isomorphisms, the [Kan–Thurston theorem](https://doi.org/10.1016/0040-9383(76)90040-9), and then apply the above.