Let $kG$ be a group algebra, and $D(kG)$ its quantum double. We first prove that the structure of the Grothendieck ring of $D(kG)$ can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of $G$. As a special case, we then give an application to the group algebra $kD_n $, where $k$ is a field of characteristic $2$ and $D_n $ is a dihedral group of order $2n$.
The structure of the unit group of the group algebra of the group $A_4$ over any finite field of characteristic 2 is established in terms of split extensions of cyclic groups.