Yetter-Drinfeld category for the quasi-Turaev group coalgebra

Let $\pi$ be a group. The aim of this paper is to construct the category of Yetter-Drinfeld modules over the quasi-Turaev group coalgebra $H=(\{H_{\text{a}}{\}}_{\text{a}\in \pi },\mathrm{\Delta },\epsilon ,S,\mathrm{\Phi })$, and prove that this category is isomorphic to the center of the representation category of $H$. Therefore a new Turaev braided group category is constructed.