On the indecomposable involutive solutions of the Yang-Baxter equation of finite primitive level

In this paper, we study the class of indecomposable involutive solutions of the Yang-Baxter equation of finite primitive level, recently introduced by Cedó and Okniński in \cite{cedo2021constructing}. We give a group-theoretic characterization of these solutions by means of displacements groups and we apply this result to compute and enumerate the ones having small size. For some classes of indecomposable involutive solutions recently studied in literature, we compute the exact value of the primitive level. Some relationships with other families of solutions also are discussed. Finally, following \cite[Question 3.2]{cedo2021constructing}, we completely describe the ones having primitive level $2$ by left braces.