Algebra, group, and Hopf Rota-Baxter operators

We know definition of Rota–Baxter operators on different algebraic systems. For examples, on groups, on algebras, on Hopf algebras. On some algebraic systems it is possible to define different types of Rota–Baxter operators. For example, on group algebra it is possible to define Rota–Baxter operator as on associative algebra, group Rota–Baxter operator and Rota–Baxter operator as on a Hopf algebra. We are investigating the following question: What are connections between these operators? We are studying these questions for the Sweedler algebra $H_4$, that is 4-dimension non–cocommutative Hopf algebra.