  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 $b$. In particular, we prove that, if a finite
group $G$ admits a diagonal double Kodaira structure, 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 $16$.
Such surfaces are different from the ones recently constructed by Lee, L\"onne
and Rollenske and, as far as we know, they provide the first examples of
positive-dimensional families of double Kodaira fibrations with small
signature.
