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 $\mathbb{Z}\mathrm{/}2$-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 $2g+2=n=2q$.