A rational cohomology which including that of a spin hyperelliptic mapping class group

Let $\mathfrak{G}=\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$ be the $\mathbb{Z}/2$-extension of the product of two symmetric groups $\mathfrak{S}_{q} \times \mathfrak{S}_q$.

In this paper, we compute the $\mathfrak{G}$-invariant part of the rational cohomology of the pure braid group $P_{n}$.

This includes the rational cohomology of a spin hyperelliptic mapping class group of genus $g$, where $2g+2=n=2q$.