Indecomposable non-degenerate 2-permutational solutions of the Yang-Baxter equation

We present a complete characterization of all indecomposable non-degenerate, not necessarily involutive, solutions of the Yang-Baxter equation of multipermutation level~2. We show that every such solution is a homomorphic image of a special, ``largest'' solution called \emph{the universal} one. On the other hand we prove that there is much simpler description. At first, on the product of a group $Z_n^2$ and an abelian group $G$, we construct some family of indecomposable non-degenerate solutions of the Yang-Baxter equation of multipermutation level $2$. Next, applying Rosenbaum's theorem of subgroups of a semidirect product and isolating a triple: a subgroup of $G$, a subgroup of $Z_n^2$ and one group homomorphism, we obtain a~full description of each epimorphism which gives the desired solutions. Such a construction provides a tool how to find (and possibly enumerate) all indecomposable non-degenerate solutions of multipermutation level $2$. We also argue that the automorphism group of the discussed solutions is regular.