Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds

For $n\ge 3$, let ${\bar{B}}_n=B_n\mathrm{/}[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup ${\bar{P}}_n=P_n\mathrm{/}[P_n,P_n]\cong {\mathbb{Z}}^N$, where $N=\left(\genfrac{}{}{0px}{}{n}{2}\right)$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some ${\bar{B}}_n$. More precisely, for every $n\ge n_M$, where $n_M\in \{3,4,5\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup ${\mathrm{\Gamma }}_n(M)\le {\bar{B}}_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\cong {\mathbb{Z}}^{N-3}$ consisting of translations such that $${\mathrm{\Gamma }}_n(M)\mathrm{/}{K}_n(M)\cong {\pi }_1(M)\mathrm{.}$$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.