  We connect properties of set-theoretic solutions to the Yang--Baxter equation
to properties of their permutation skew brace. In particular, a variation of
the multipermutation level of a solution is presented and we show that it
coincides with the multipermutation level of the permutation skew brace,
contrary to the inequality that one has for the usual multipermutation level of
solutions. We relate the number of orbits of a solution to generators of its
permutation skew brace and relate different kinds of notions of generating sets
of a skew brace. Also, the automorphism groups of solutions are studied through
their permutation skew brace. As an application, we obtain a surprising result
on subsolutions of multipermutation solutions and we give a description of all
finite indecomposable involutive solutions to the Yang--Baxter equation with
abelian permutation group. For multipermutation level 3, we obtain the precise
number of isomorphism classes of such solutions of a given size.
