Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

Let be a compact Riemann surface of genus b. We investigate finite quotients G of the pure braid group on two strands which do not factor through . 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 , and we completely classify the cases where 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.