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EDIT:I am not sure, whether I understand the second paragraph correctly. Group homology can be defined as the homology opf the classifying space, which reads $H_(G):=H_(BG)$. So this causes confusion, if it is not clear, whether one considers the group homology or the homology of the discrete space given by forgetting the group structure.

If the second was the case: Then the answer would be: write the discrete space as a union of orbits $\amalg_{i\in I}G/H_i$ and the result would be (like above) $\bigoplus_{i\in I} H_*(H_i)$ (denoting the group homology).

Or (more likely) it asks for some sort of "equivariant group homology", which I would read as $H_*^C_p(BC_q)$. Using a nice functorial construction for $BC_q$, the $C_p$ action on the group also gives a $C_p$ action on the classifying space $BC_q$ and also on $EC_q$. However in this case I would guess, that $EC_q\times EC_p$ is a free,contractible $C_p\ltimes C_q$-CW-complex. Then it would be a model for $EC_p\ltimes C_q$ and hence:

$(BC_q \times EC_p)/C_p=(EC_q/C_q\times EC_p)/C_p=(EC_q\times EC_p)/(C_p\ltimes C_q)$.Then $H_*^{C_p}(C_q)$ would just be the group homology of $C_p\ltimes C_q$.

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Sorry this got too long for a comment, so I post it as an answer.

A CW-complex equipped with a $G$- action is not the right thing to consider. A $G$-CW complex is a space, that can be built using blocks of the form $G/H \times (D^n,S^{n-1})$ (in the same way you build a CW-complex). A $G$-CW complex has better properties than a CW-complex, for example the fixed point set of any subgroup is a subcomplex. However most CW-complexes with a $G$ action can also be given the structure of a CW-complex.

In the example of $G=\{1,t\}=\mathbb{Z}/2$ acting on $S^1\times S^1$ by flipping the components, one could for example take the following $G$-CW- structure with 1 $0$-cell $P$ of type $G/G$, two 1-cells $A,B$ of type $G/1,G/G$ and 1 2-cell $C$ of type $G/1$:

Then the cellular chain complex may be considered as $\mathbb{Z}[G]$ chain complex. Here it is:

$\mathbb{Z}[P]\leftarrow\mathbb{Z}[A,tA,B] \leftarrow \mathbb{Z}[C]$

, where the differentials are given by $A,B\mapsto 0, C\mapsto A+tA+B$. Forgetting the names, we can write the chain complex as

$\mathbb{Z}[G/G]\leftarrow\mathbb{Z}[G/1]\oplus \mathbb{Z}[G/G] \leftarrow \mathbb{Z}[G/1]$

These maps are $G$-equivariant. If one would apply $\otimes_{\mathbb{Z}[G]}\mathbb{Z}$ to this chain complex and take the homology, one would just get the cellular homology of the quotient $H_*(X/G)$.

I think, that if one takes a projective resolution of this chain complex first, one should get $H_*(X\times_GEG)$. (The cellular chain complex of $X\times EG$ is a free resolution of the cellular chain complex of $X$).