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Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given by the isomorphism of $\pi$ with $\text{Aut}(\overline{X})$. Let $A$ be an abelian group and let $\mathbb{Z}[\pi]$ act trivially on $A$, $a \cdot \sigma = a$ for $\sigma \in \pi$ and $a \in A$.

Now, assume that $X$ is a CW complex. How do I see that $\overline{X}$ is a CW complex such that the action of $\pi$ on $\overline{X}$ induces an action of the group ring $\mathbb{Z}[\pi]$ on the cellular chain complex $C_*(\overline{X})$ such that each $C_q(\overline{X})$ is a free $\mathbb{Z}[\pi]$-module and$$C_*(X; A) \cong A \otimes_{\mathbb{Z}[\pi]} C_*(\overline{X})?$$

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    $\begingroup$ The deck transformations take cells to cells. That's the action of $\pi$ on $C_*(\overline X)$. (You put a cell structure on $\overline X$ such that there are $|\pi |$-many cells lying above a cell in $X$.) $\endgroup$ – user98602 Nov 3 '15 at 17:37
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You really need to take a CW structure on $\bar X$ which is inherited by a CW structure on $X$. This means a priori your CW structure is fixed during the whole construction, and in the end you show the independence of the resulting homology of the chosen structure (as you do in cellular homology as well).

Fix a CW structure on $X$. The covering space $p:\bar X\to X$ inherits that structure by defining a cellular structure by choosing all lifts of all cells as decomposition. Note that the preimage of a cell $p^{-1}\sigma$ is canonically (by choosing basepoint) homeomorphic to $\pi \times \sigma$ with $\pi$ acting on those $\pi$-many copies of $\sigma$ just by multiplication.

So $C_*(X)$ is just the free $\mathbb Z$-module generated by the cells of $X$ and similarly $C_*(\bar X)$ is (as a group ring module) the free $\mathbb Z\pi$-module generated by the (lift at the basepoint of the) cells in $X$.

This leads to the definition of twisted homology or homology with local coefficients. Namely given a left-$\pi$-module $A$ (one has to be a little careful in the nonabelian context distinguishing left and right modules, but with the natural involution of the group ring we can survive those issues) one can compute homology with $A$ as coefficients by taking $H_*(X;A)= H_*(C_\bullet (\bar X) \otimes_\pi A )$. One shows that this is independent of the chosen cellular structure.

So $A$ being an $\pi$-module means having a rep $\rho : \pi \to Aut(A)$. You can see that it suffices to look at the intermediate cover corresponding to $ker \rho$ (which also has a cellular $\mathbb Z[ \pi/ker\rho]$ chain complex) and twist the coefficients there instead of doing this in the universal cover (this can be for example found in Hatcher's "Algebraic Topology" book which gives a short intro to this). This means that for your question, where the action factors through augmentation, one notes:

$$ C_\bullet(\bar X) \otimes_\pi A = C_\bullet (X) \otimes_{\mathbb Z} A = C_\bullet(X;A). $$

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