Every group is the automorphism group of a set-theoretic solution of the Yang--Baxter equation

We give an affirmative answer to the abstract group problem for the category of all set-theoretic solutions of the Yang--Baxter (YB) equation: every group is isomorphic to the automorphism group of a YB-solution, which can be chosen on a finite set if the group is finite. In the terminology of László Babai, this means that the category is universal and finitely universal, which is an analogue of Frucht's classical theorem for graphs. Furthermore, the YB-solutions can be chosen within three very small classes: the solutions of the Frobenius-Separability equation, the solutions of the Baaj--Long--Skandalis equation and the recently introduced bipartite solutions. The proof is based on new constructions which associate to any graph a YB-solution in each of these classes, with the same automorphism group as the graph. Explicit YB-solutions with automorphism groups Cₙ, Sₙ, Dₙ and Q₈ are given.