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 $\Bbbk^G {}^\tau \#_{\sigma} \Bbbk F$, constructed as abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for a finite group $G$ and an arbitrary group $F$. We investigate when a coquasitriangular structure exists on $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ and provide characterizations of its coquasitriangular structures. As an application, we study the coquasitriangular structures for the case where $G = \mathbb{Z}_2$.