# Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology $$\alpha_n: C(\mathbb{R}^n;X)\to \Omega^n\Sigma^n X.$$ Moreover, I find that $$\Omega^n\Sigma^nS^0=\Omega^n S^n=\bigsqcup_{q\in \mathbb{Z}}\{f\in [S^n,S^n]_*\mid \deg (f)=q\}.$$ Let $$\{f\in [S^n,S^n]_*\mid \deg (f)=q\}=\Omega_q^nS^n$$ then $$\Omega^n\Sigma^nS^0=\bigsqcup_{q\in\mathbb{Z}}\Omega_q^nS^n.$$ I find in some references that $$C(\mathbb{R}^n;S^0)=\bigsqcup_{k\geq 0}F(\mathbb{R}^n,k)/\Sigma_k.$$ Let $\alpha_{n,k}$ be the restriction of $\alpha_n$ to $F(\mathbb{R}^n,k)/\Sigma_k$.

Question: I want to obtain the ring structure of $H^*(F(\mathbb{R}^n,k)/\Sigma_k;\mathbb{Z}_2)$. Is there any way to obtain it? Do we have a well-defined map $$\alpha_{n,k}: F(\mathbb{R}^n,k)/\Sigma_k\to\Omega_k^nS^n$$ that induces a ring isomorphism $$H^*(\Omega_k^nS^n;\mathbb{Z}_2)\to H^*(F(\mathbb{R}^n,k)/\Sigma_k;\mathbb{Z}_2)?$$

There is a a well-defined map from $F(\mathbb R^n,k)/\Sigma_k \to \Omega^n_k S^n$ but it only induces an iso on $H^*$ for $* \leq k/2$.