  A new family of non-degenerate involutive set-theoretic solutions of the
Yang-Baxter equation is constructed. All these solutions are strong twisted
unions of multipermutation solutions of multipermutation level at most two. A
large subfamily consists of irretractable and square-free solutions. This
subfamily includes a recent example of Vendramin, who first gave a
counterexample to Gateva-Ivanova's Strong Conjecture. All the solutions in this
subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and
also they answer a question of Cameron and Gateva-Ivanova. It is proved that
the natural left brace structure on the permutation group of the solutions in
this family has trivial socle. Properties of the permutation group and of the
structure group associated to these solutions are also investigated. In
particular, it is proved that the structure groups of finite solutions in this
subfamily are not poly-(infinite cyclic) groups.
