Coquasitriangular structures on Hopf algebras constructed via abelian extensions

The aim of this paper is to study coquasitriangular structures on a class of cosemisimple Hopf algebras of the form ${\mathbb{k}}^G{}^{\tau }{\mathrm{\#}}_{\sigma }\mathbb{k}F$, constructed as abelian extensions of $\mathbb{k}F$ by ${\mathbb{k}}^G$ for a finite group $G$ and an arbitrary group $F$. We investigate when a coquasitriangular structure exists on ${\mathbb{k}}^G{}^{\tau }{\mathrm{\#}}_{\sigma }\mathbb{k}F$ and provide characterizations of its coquasitriangular structures. As an application, we study the coquasitriangular structures for the case where $G={\mathbb{Z}}_2$.