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

Yetter-Drinfeld modules for group-cograded Hopf quasigroups