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 $(G,\mathrm{\Psi })$ defines a Rota–Baxter operator on the semi-direct product $H{⋊}_{\mathrm{\Psi }}G$. On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product $H{⋊}_{\mathrm{\Psi }}G$ defines a relative Rota–Baxter operator on $H$ with respect to $(G,\mathrm{\Psi })$. We introduce homomorphic post-groups and find their connection with $\lambda$-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.