Some properties of relative Rota--Baxter operators on groups

We find connection between relative Rota--Baxter operators and usual Rota--Baxter operators. We prove that any relative Rota--Baxter operator on a group H with respect to (G, Ψ) defines a Rota--Baxter operator on the semi-direct product HΨ G. On the other side, we give condition under which a Rota--Baxter operator on the semi-direct product HΨ G defines a relative Rota--Baxter operator on H with respect to (G, Ψ). We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.