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We connect properties of set-theoretic solutions to the Yang--Baxter equationto properties of their permutation skew brace. In particular, a variation ofthe multipermutation level of a solution is presented and we show that itcoincides with the multipermutation level of the permutation skew brace,contrary to the inequality that one has for the usual multipermutation level ofsolutions. We relate the number of orbits of a solution to generators of itspermutation skew brace and relate different kinds of notions of generating setsof a skew brace. Also, the automorphism groups of solutions are studied throughtheir permutation skew brace. As an application, we obtain a surprising resulton subsolutions of multipermutation solutions and we give a description of allfinite indecomposable involutive solutions to the Yang--Baxter equation withabelian permutation group. For multipermutation level 3, we obtain the precisenumber of isomorphism classes of such solutions of a given size.

Studying solutions of the Yang-Baxter equation through skew braces, with  an application to indecomposable involutive solutions with abelian  permutation group