The Batalin-Vilkovisky structure on the Tate-Hochschild cohomology ring of a group algebra

We determine the Batalin-Vilkovisky structure on the Tate-Hochschild cohomology of the group algebra $kG$ of a finite group $G$ in terms of the additive decomposition. In particular, we show that the Tate cohomology of $G$ is a Batalin-Vilkovisky subalgebra of the Tate-Hochschild cohomology of the group algebra $kG$, and that the Tate cochain complex of $G$ is a cyclic $A_{\infty}$-subalgebra of the Tate-Hochschild cochain complex of $kG$.