Let $X$ be a pointed topological space, $k$ a field (or I suppose, more generally a commutative ring) and $n$ a positive integer.
We know (by a reduced version of the Eilenberg-Zilber theorem) that there is an equivalence (=quasi-isomorphism)
$$\tilde{C}_{*}(X^{\land n};k) \to \tilde{C}_{*}(X;k)^{\otimes n} $$
Both objects in the above carry a natural $\Sigma_n$-action. My question is - (when) does this give an equivalence
$$\tilde C_*(X^{\land n}_{\Sigma_n};k) \to \tilde{C}_*(X;k)^{\otimes n}_{\Sigma_n}?$$
(or perhaps more poetically $\tilde{C}_*(\mathrm{Sym}^n(X);k) \simeq \mathrm{Sym}^n(\tilde{C}_*(X;k))$)
We can certainly take homotopy $\Sigma_n$-coinvariants and I believe this gives the correct object on the right-hand side (due to cofibrance of the objects) but I'm less sure on what is obtained on the left.
Looking at related questions such as this, this and this, it seems the equivalence ought to hold over a field of characteristic $0$; I'm particularly interested in the positive characteristic case and with what goes wrong.
In any case I'd be grateful for any reference for the compatibility of actions on spaces with corresponding actions on (co)homology! Thanks.
