This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for computing the equivariant cohomology of a space, and asked how they extend to generalized cohomology. However, it now occurs to me that when the space is a manifold, there is a third spectral sequence for computing equivariant cohomology, and that for my application this is the one I am really interested in.
Let $G$ be a group with sufficiently smooth action on $X$, a manifold of dimension $d$. For an Abelian group $A$, let $\mathcal{C}_n(X,A)$ be the group of $n$-chains (not cochains) on $X$ with $A$ coefficients. We can treat this as a $G$-module (Abelian group with compatible $G$-action) with the $G$-action inherited from the $G$-action on $X$.
Now for any $G$-module $B$, we can introduce the Abelian group of "group cochains" $\mathcal{C}^m(G, B)$ that are used to define group cohomology. For example, we can define $\mathcal{C}^m(G, B) = \mathrm{Hom}_G(F_n, B)$, where $$\cdots F_n \to F_{n-1} \to \cdots \to F_0 \to \mathbb{Z} \to 0$$ is a projective $\mathbb{Z}[G]$-resolution of the integers.
In particular, $\widetilde{Q}_{m,n} := \mathcal{C}^m(G, \mathcal{C}_{d-n}(X,A))$ defines a double complex. From Oscar Randal-Williams' very nice answer to my question
Is there a kind of Poincare duality for Borel equivariant cohomology?
we know that the total cohomology of this double complex is the equivariant cohomology $\mathcal{H}^{\bullet}( (X \times EG)/G, A)$ (this would be immediate if we were talking about cochains on $X$, but since we are talking about chains we have to invoke Poincare duality as discussed there).
We can therefore compute the equivariant cohomology by a spectral sequence. There are two possible spectral sequences for a double complex: I am interested in the one where we take the $d_1$ differentials to be the coboundary on $G$-cochains.
I do not think, however, that this spectral sequence is equivalent to the isotropy spectral sequence (which is the Leray spectral sequence for the map $X \to X/G$, and is what one would get if one replaced $X$ chains with co-chains in the double complex).
So now I would like to ask whether, and how, this mysterious third spectral sequence extends to computing the generalized equivariant cohomology $\mathcal{F}^{\bullet}( (X \times EG)/G, A)$ for some generalized cohomology theory $\mathcal{F}^{\bullet}$?
