  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
$H=(\{H_\a\}_{\a\in\pi},\Delta,\varepsilon,S,\Phi)$, 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.
