Group extensions, fiber bundles, and a parametric Yang–Baxter equation

We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal bundles and that the case of vector bundles considered earlier is an infinitesimal version of the case of a solution coming from the principal bundle structure.