  Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the
notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is
bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of
Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the
subcategory $\mathscr Y\mathscr D(H)$ of Yetter-Drinfeld modules is a braided
crossed category.
