On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^{\ast }$ of Kashina

Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong {⨂}_{i\in I}\mathcal{B}(N_i)$, where $N={⨁}_{i\in I}{N}_i$, each $N_i$ is a simple object in ${}_H^H\mathcal{Y}\mathcal{D}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal{B}(N)$.