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 $(G, \Psi)$ defines a Rota--Baxter operator on the semi-direct product $H\rtimes_{\Psi} G$. On the other side, we give condition under which a Rota--Baxter operator on the semi-direct product $H\rtimes_{\Psi} G$ defines a relative Rota--Baxter operator on $H$ with respect to $(G, \Psi)$. 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.