Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-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, $\mathcal{C}^m(G, \mathcal{C}^n(X,A))$ defines a double complex. Now for any double complex there are two spectral sequences we can use to compute the total cohomology. In this case, the total cohomology is equivalent to the equivariant cohomology $H^{\bullet}_G(X,A) = H^{\bullet}(X//G,A)$. One of the two spectral sequences can be identified as the Serre spectral sequence associated with the fibration $X \to X//G \to BG$. The other one does not appear to have a name.

My question now is, suppose that we now replace ordinary cohomology with generalized cohomology, that is, we want to compute $\hat{H}^{\bullet}(X//G)$ for some generalized cohomology theory $\hat{H}$. The Serre spectral sequence generalizes to the Atiyah-Hirzebruch spectral sequence. But does the other, unnamed, spectral sequence have any analogous generalization?