The theory of Rota--Baxter operators on rings and algebras has been developed since 1960. In 2020, the notion of Rota--Baxter operator on a group was defined. Further, it was proved that one may define a skew left brace on any group endowed with a Rota--Baxter operator. Thus, a group endowed with a Rota--Baxter operator gives rise to a set-theoretical solution to the Yang--Baxter equation. We provide some general constructions of Rota--Baxter operators on a group. Given a map on a group, we study its extensions to a Rota--Baxter operator. We state the connection between Rota--Baxter operators on a group and Rota--Baxter operators on an associated Lie ring. We describe Rota--Baxter operators on sporadic simple groups.