Let \(\Sigma _b\) be a compact Riemann surface of genus b. We investigate finite quotients G of the pure braid group on two strands \(\textsf{P}_2(\Sigma _b)\) which do not factor through \(\pi _1(\Sigma _b \times \Sigma _b)\) . Building on our previous work on some special systems of generators on finite groups that we called diagonal double Kodaira structures, we prove that, if G does not have order 32, then \(|G| \ge 64\) , and we completely classify the cases where \(b=2\) and equality holds. In the last section, as a geometric application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group.
