Group cohomology with coefficients in a chain complex Let us suppose that I'm in the following situation: I have a chain complex $(C,\partial)$ and say a finite group $G$ acting over $C$ up to homotopy, meaning that for each  $g \in G$ I have a self chain homotopy equivalence $g_\#:C\to C$ such that $h_\# \circ g_\#\simeq(f\circ g)_\#$, where $\simeq$ is a chain homotopy.  
If $\partial=0$ the situation would collapse to the one of a group acting over $C$ (simply a module in this case) and I could consider the group cohomology of $G$ with coefficients in $C$. So my question is:

Is there a meaningful notion of cohomology of $G$ with coefficients in a chain complex? 

PS: of course I can take the homology and get a proper action of $G$ on $H_*(C)$ but I find this boring somehow.
 A: Given your interest in Floer theory, I can't resist mentioning recent work of Hendricks, Lipshitz, and Sarkar, where they describe the equivariant homology of a chain complex with a coherent group action, and apply it to define equivariant Floer homology without the need for a $G$-invariant perturbation. 
To describe what a coherent group action is, consider the following picture: you have an "involution" $\iota$ on the chain complex, so that $\iota^2$ is not the identity, but is homotopic to the identity. Record this homotopy as $H: \iota^2 \to 1$. Then you will not have $\iota H = H \iota$, but these might further be related by a homotopy, $H_2$, ... ultimately you will have constructed an infinite sequence $H_n$ of homotopies which satisfy various relations with $\iota$ and one another.
Their first paper is here, and describes the story for a finite group $G$; the second paper is here, and gives the story for a compact Lie group $G$. 
Personally, I find the second paper an easier read: it uses the notion of "homotopy colimit of an $\infty$-category", but everything is very explicit, and they are patient in explaining to the reader what's going on - I found it a joy to read along and learn from. (It also better separates the Floer theory from the homological algebra.) If you'd like to specialize to the finite group case, you can compare to the area around 3.8-3.11 in the first paper.

If the $G$-action is strict, this is an easier story. You can either use the bar construction $B(\Bbb Z, \Bbb Z[G], C)$ to obtain a chain complex whose homology is the equivariant homology of $C$, or you could pick a resolution of $C$ by some half-plane bicomplex (where $C$ lives on the $x = -1$ line) and totalize it. (The bar construction is an example of this.) 
In all cases there is a spectral sequence $$H_*(G;H_*(C)) \implies H_*(G;C).$$ 
The discussion for compact Lie groups is (as always) more complicated: instead of the algebra $\Bbb Z[G]$, one must take homology with respect to the dg-algebra $C_*(G;\Bbb Z)$. I won't discuss this unless you really want to know about it. 
