LAUSR

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 $kG$.