  In this paper, we study the class of indecomposable involutive solutions of
the Yang-Baxter equation of finite primitive level, recently introduced by
Ced\'o and Okni\'nski 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.
