Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on $$ \prod_n X $$ by $$ \sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in \Sigma_n. $$ Then we have a quotient space $$ (\prod_n X)/\Sigma_n. $$ By Kunneth formula, $$ H^*(\prod_nX; F)=\bigotimes_nH^* ( X; F). $$

**Question:** does
$$
H^*((\prod_n X)/\Sigma_n; F)=(\bigotimes_nH^* ( X; F))/\Sigma_n=\text{Symmetric tensor product }_nH^* ( X; F)
$$
or not? **For example, how to compute
$$
H^*((\prod_n S^m)/\Sigma_n;\mathbb{Z}_2)
$$
explicitly?**