Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients

Let $\mathsf{B}_2(\Sigma_g)$ be the full braid group on two strings on a compact Riemann surface of genus $g$. We compute the number of finite cyclic, dihedral and extra-special quotients $\varphi \colon \mathsf{B}_2(\Sigma_g) \to G$, under the assumption that the quotient map $\varphi$ does not factor through $\pi_1(\operatorname{Sym^2}\Sigma_g)$. We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of $\operatorname{Sym^2}(\Sigma_g)$ branched on the diagonal. In particular, we construct two $3$-dimensional families of minimal surfaces of general type with $p_g=7$, $q=4$ and $K^2=32$ such that members of different families have the same biregular invariants and the same Betti numbers, but different torsion part for the first homology group.