Some classifications of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:x^{2}y}$ of Kashina

Let $H$ be the $16$-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:x^{2}y}$ classified by Kashina. We figure out all simple Yetter-Drinfeld $H$-modules, and then determine all finite-dimensional Nichols algebras satisfying the constraint condition $\mathcal{B}(V)\cong {⨂}_{i\in I}\mathcal{B}(V_i)$, where $V={⨁}_{i\in I}{V}_i$, each $V_i$ is a simple object in ${}_H^H\mathcal{Y}\mathcal{D}$. Finally, we describe some liftings of the corresponding Radford biproducts $\mathcal{B}(V)\mathrm{♯}H$, which provide some classifications of finite dimensional Hopf algebras with $H$ as their coradical.