Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258n, the cohomology ring $$ H^*(F(\mathbb{R}P^n,k);R)$$ is obtained for any commutative ring $R$ with unit and $2$ invertible. I want to find $$ H^*(F(\mathbb{R}P^n,2);\mathbb{Z}).$$

When I use Serre spectral sequence for fibration $$ \mathbb{R}P^n\setminus *\simeq \mathbb{R}P^{n-1}\to F(\mathbb{R}P^n,2)\to \mathbb{R}P^n, $$ I do not know how to determine the differentials. How to get $$ H^*(F(\mathbb{R}P^n,2);\mathbb{Z})?$$

Is there general result for $$ H^*(F(\mathbb{R}P^n,k);\mathbb{Z})?$$

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring $$ H^*(F(\mathbb{R}P^n,k);R)$$ is obtained for any commutative ring $R$ with unit and $2$ invertible. I want to find $$ H^*(F(\mathbb{R}P^n,2);\mathbb{Z}).$$

When I use the Serre spectral sequence for the fibration $$ \mathbb{R}P^n\setminus *\simeq \mathbb{R}P^{n-1}\to F(\mathbb{R}P^n,2)\to \mathbb{R}P^n, $$ I do not know how to determine the differentials. How can I get $$ H^*(F(\mathbb{R}P^n,2);\mathbb{Z})?$$

Are there more general results for $$ H^*(F(\mathbb{R}P^n,k);\mathbb{Z})?$$