The Tate-Hochschild cohomology ring of a group algebra

We show that the Tate-Hochschild cohomology ring $H{H}^{\ast }(RG,RG)$ of a finite group algebra $RG$ is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of $G$. Moreover, our main result provides an explicit formula for the cup product in $H{H}^{\ast }(RG,RG)$ with respect to this decomposition. As an example, this formula helps us to compute the Tate-Hochschild cohomology ring of the symmetric group $S_3$ with coefficients in a field of characteristic 3.