  We exhibit new examples of double Kodaira fibrations by using finite Galois
covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth
projective curve of genus $b \geq 2$. Each cover is obtained by providing an
explicit group epimorphism from the pure braid group $\mathsf{P}_2(\Sigma_b)$
to some finite Heisenberg group. In this way, we are able to show that every
curve of genus $b$ is the base of a double Kodaira fibration; moreover, the
number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed
curve $\Sigma_b$ is at least $\boldsymbol{\omega}(b+1)$, where
$\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for the
arithmetic function counting the number of distinct prime factors of a positive
integer. As a particular case of our general construction, we obtain a real
$4$-manifold of signature $144$ that can be realized as a real surface bundle
over a surface of genus $2$, with fibre genus $325$, in two different ways.
This provides (to our knowledge) the first "double" solution to a problem from
Kirby's list in low-dimensional topology.
