Braid groups, mapping class groups and their homology with twisted coefficients

We consider the Birman-Hilden inclusion $\phi \text{\hspace{0.17em}⁣}:{\mathfrak{B}\mathfrak{r}}_{2g+1}\to {\mathrm{\Gamma }}_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\phi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1({\mathrm{\Sigma }}_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in ${\phi }^{\ast }(H_1({\mathrm{\Sigma }}_{g,1}))$ has only $4$-torsion.