Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina

Let $H$ be the 16-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:1}$ appeared in Kashina’s work \cite{K00}. We obtain all simple Yetter-Drinfeld modules over $H$ and then determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(V)\cong {⨂}_{i\in I}\mathcal{B}(V_i)$, where $V={⨁}_{i\in I}{V}_i$, each $V_i$ is a simple object in ${}_H^H\mathcal{Y}\mathcal{D}$. Finally, we describe all liftings of those $\mathcal{B}(V)$ over $H$.