For a group $\pi$, the objective of this paper is to construct a class of quasitriangular Hopf $\pi$-coalgebra. We first shall present the new tool called a group unified coprocut, followed by a classification result for $\pi$-unified coproducts in virtue of an algebra lazy 1-$\pi$-cycle which is the dual to that defined by Bichon and Kassel. Then, we discuss when a $\pi$-unified coproduct has a quasitriangular structure. Finally, some applications of our main results are considered.