On a class of involutive Yang-Baxter groups

A group is called an involutive Yang-Baxter group (IYB-group) if it is isomorphic to the permutation group of an involutive, non-degenerate set-theoretic solution of the Yang-Baxter equation. This paper investigates finite soluble groups whose Sylow subgroups have nilpotency class at most two, addressing Cedó and Okniński’s question~\cite{CedoOkninski2025} of whether such groups are IYB-groups. We establish that a finite soluble group with Sylow subgroups of class at most two is an IYB-group if its nilpotent residual is ${\text{Q}}_8$-free. We also prove that a finite soluble group with Sylow subgroups of class at most two and Sylow $2$-subgroups isomorphic to $Q_8$ is an IYB-group.