Yetter-Drinfeld modules for group-cograded Hopf quasigroups

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 $\mathcal{Y}\mathcal{D}\mathcal{Q}(H)$ of Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the subcategory $\mathcal{Y}\mathcal{D}(H)$ of Yetter-Drinfeld modules is a braided crossed category.