A class of (infinite-dimensional) cosemisimple Hopf algebras constructed via abelian extensions

In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra ${\mathbb{k}}^G{}^{\tau }{\mathrm{\#}}_{\sigma }\mathbb{k}F$ constructed through abelian extensions of $\mathbb{k}F$ by ${\mathbb{k}}^G$ for some (infinite) group $F$ and finite group $G$ is cosemisimple, and discuss when it admits a compact quantum group structure if $\mathbb{k}$ is the field of complex numbers $\mathbb{C}\mathrm{.}$ We also find all the simple ${\mathbb{k}}^G{}^{\tau }{\mathrm{\#}}_{\sigma }\mathbb{k}F$-comodules and attempt to determine the Grothendieck ring of the category of finite-dimensional right ${\mathbb{k}}^G{}^{\tau }{\mathrm{\#}}_{\sigma }\mathbb{k}F$-comodules. Moreover, some new properties are given and some new examples are constructed.