Homological stability of spin mapping class groups and quadratic symplectic groups

We study the homological stability of spin mapping class groups of surfaces and of quadratic symplectic groups using cellular $E_2$-algebras. We get improvements in their stability results, which for the spin mapping class groups we show to be optimal away from the prime $2$. We also prove that in both cases the $\mathbb{F}_2$-homology satisfies secondary homological stability. Finally, we give full descriptions of the first homology groups of the spin mapping class groups and of the quadratic symplectic groups.