TITLE: Equivariant cohomology vs. invariant cohomology vs. cohomology of quotient space
QUESTION [16 upvotes]: Given a space $X$ and an action of a group $G$ on $X$, the $G$-invariant cochains with coefficients in an Abelian group $A$ define a sub-cocomplex $\mathcal{C}^{\bullet}_G$ of the cocomplex $\mathcal{C}^{\bullet}$ of cochains with coefficients in $A$. I will call the cohomology $H^{\bullet}(\mathcal{C}_G)$ the "invariant cohomology" of $X$. What is the relationship between the follwing three objects: (i) the invariant cohomology, (ii) the usual equivariant cohomology $\mathcal{H}_G^{\bullet}(X, A)$, and (iii) the cohomology of the quotient space $H^{\bullet}(X/G,A)$? Note that I am most interested in the case where $A$ is a finite Abelian group.
I think that there are always homomorphisms
$$H^n(X/G,A) \to H^n(\mathcal{C}_G) \to \mathcal{H}_G^n(X,A).$$
The first map comes from pulling back the projection $X \to X/G$, and the second can be seen from the interpretation of equivariant cohomology as the total cohomology of the double cocomplex of group cochains of $G$ valued in $\mathcal{C}^{\bullet}$. 
But how does one characterize the kernel and image of these maps? Is there a simple statement if $X$ is contractible, say?
[Edit: regarding the relationship between $H^n(X/G,A)$ and $H^n(\mathcal{C}_G)$, please see the comments on Mark Grant's answer. The upshot is that with simplicial cochains, $H^n(\mathcal{C}_G)$ seems to depend on the choice of triangulation. But, if one defines $\mathcal{C}_G$ in a suitably triangulation-independent way, i.e. either singular cochains, or simplicial cochains on the abstract simplicial complex containing all simplex embeddings into X, then $H^n(\mathcal{C}_G)$ and $H^n(X/G,A)$ are almost certainly not the same.]

REPLY [4 votes]: If $G$ is a finite group acting by simplicial automorphisms on a simplicial complex $X$, then 
$$H^{\ast}(X/G;\mathbb{Q}) = (H^{\ast}(X;\mathbb{Q}))^{G} = H^{\ast}((C^{\ast}(X;\mathbb{Q}))^G),$$
where the superscripts indicate that we are taking the invariants with respect to $G$.  For a proof, see Proposition 1.1 (and its proof) in my note "The action on homology of finite groups of automorphisms of surfaces and graphs", available on my webpage here.  It is definitely necessary to take $\mathbb{Q}$ coefficients here (the referenced proof will make it clear why this is the case).