Abstract: Let G be a group, using the properties of Turaev braided group categories, we constructed a Turaev braided Gcategory over DoiHopf datum (H,A,C), where H,A,C were Hopf algebras. Furthermore, when C was finitedimensional Hopf algebra, we constructed a quasitriangular Turaev Gcoalgebra A#^H C* over a family of Smash product algebras {A#^HC*(α)}α∈G whose representation category and AM^C(H) were isomorphic.

Key words: Turaev braided G-category, Turaev G-coalgebra, Doi-Hopf module