  We study some special systems of generators on finite groups, introduced in
previous work by the first author and called "diagonal double Kodaira
structures", in order to investigate non-abelian, finite quotients of the pure
braid group on two strands $\mathsf{P}_2(\Sigma_b)$, where $\Sigma_b$ is a
closed Riemann surface of genus $b$. Our main result is that, if $G$ is such a
quotient, then $|G|\geq 32$, and equality holds if and only if $G$ is
extra-special. In the last section, as a geometrical application of our
algebraic results, we construct two $3$-dimensional families of double Kodaira
fibrations having signature $16$.
