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

Let ${\mathrm{\Sigma }}_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands ${\mathsf{P}}_2({\mathrm{\Sigma }}_b)$ which do not factor through ${\pi }_1({\mathrm{\Sigma }}_b\times {\mathrm{\Sigma }}_b)$. Building on our previous work on some special systems of generators on finite groups that we called \emph{diagonal double Kodaira structures}, we prove that, if $G$ has not order $32$, then $\mathrm{\mid }G\mathrm{\mid }\ge 64$, and we completely classify the cases where 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.