Symmetric multiplication on homotopy quotient of topological space According to http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf. , if $X$ is any topological space, then its cochain complex $C^* := C^*(X)$ has a (homotopy) symmetric multiplication 
$$D_2(C^*) := (C^* \otimes C^* \otimes E(\mathbb{Z}/2))_{\mathbb{Z}/2} \rightarrow C^*$$
where $E(\mathbb{Z}/2)$ is a homological model for a contractible space with free $\mathbb{Z}/2$-action. 
pre-Question: how can I describe concretely this symmetric multiplication? (For instance, if I pick a cellular model for $X$ then how do I write it down?) 
If $X$ is a topological space with $G$-action, then we can form its homotopy quotient $X/G := (X \times EG)/G$, where $EG$ is a contractible space with free $G$-action. At the level of cochains, we have $C^*(X/G) = C^*(X)^{hG} := Hom(C_*(EG), X)^G$. So I should get a symmetric multiplication 
$$ D_2(C^{hG}) \rightarrow C^{hG}$$
Question: more generally, if $C$ is any chain complex with symmetric multiplication $D_2(C) \rightarrow C$, is there an induced multiplication $D_2(C^{hG}) \rightarrow C^{hG}$? What is it? 
 A: The construction of the map 
$$D_2(C^*)\rightarrow C^*$$
goes back at least to N. E. Steenrod. In his beautiful paper 
"Products  of  cocycles  and  extensions  of  mappings", Ann. of Math. (2) 48 (1947), 290–320, he gives a very explicit combinatorial description of this map.
Take for $E\mathbb{Z}/2$ the chain complex whose i-chains ($i\geq 0$) are generated by the regular reprensation of the symmetric group $\mathbb{Z}/2=\{e,\tau\}$ i.e we have $$(E\mathbb{Z}/2)_i=\mathbb{Z}<e_i,\tau.e_i>$$
together with $de_i=e_{i-1}+(-1)^{i}\tau.e_{i-1}$.
$e_i$ corresponds to the $i$-th cup-i product. N. E. Steenrod gives in his paper an explicit formula for the action of the cup-i product on cochains. 
This formulas can be found in many papers and many works where they were extended to an $E_{\infty}$-structure on singular cochains. One can cite:


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*C. Berger and B. Fresse,"Combinatorial operad actions on cochains", Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 135–174.

*J. P. May: "A general algebraic approach to Steenrod operations", The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231.

*J. McClure and J. Smith : "Multivariable cochain operations and little n-cubes", preprint arXiv:math.QA/0106024(2001).  
