On finite quotients of surface braid groups having order at most $127$

Let $\Sigma_b$ be a compact Riemann surface of genus $b \geq 2$ and let $\mathsf{P}_2(\Sigma_b)=\pi_1(\Sigma_b \times \Sigma_b - \Delta)$ be the corresponding pure braid group on two strands. A finite quotient $\varphi \colon \mathsf{P}_2(\Sigma_b) \to G$ is called "admissible" if $\varphi$ does not factor through $\pi_1(\Sigma_b \times \Sigma_b)$. In this work we classify all admissible quotients of $\mathsf{P}_2(\Sigma_b)$ such that $|G| \leq 127$.