Splitting of the homology of the punctured mapping class group

Let ${\mathrm{\Gamma }}_{g,1}^m$ be the mapping class group of the orientable surface ${\mathrm{\Sigma }}_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let ${\beta }_m({\mathrm{\Sigma }}_{g,1})$ be the braid group on $m$ strands of the surface ${\mathrm{\Sigma }}_{g,1}$. We prove that $H_(\Gamma_{g,1}^m;$\mathbb{Z}$2)\cong H(\Gamma_{g,1};H_*(\beta_{m}(\Sigma_{g,1});$\mathbb{Z}$_2))$. The main ingredient is the computation of $H_{\ast }({\beta }_m({\mathrm{\Sigma }}_{g,1});{\mathbb{Z}}_2)$ as a symplectic representation of ${\mathrm{\Gamma }}_{g,1}$.