Finite group actions on dg categories and Hochschild homology

We prove a decomposition of the Hochschild homology groups of the equivariant dg category ${\mathcal{C}}^G$ associated to a small dg category $\mathcal{C}$ with direct sums on which a finite group $G$ acts. When the ground field is $\mathbb{C}$ this decomposition is related to a categorical action of $\text{Rep}(G)$ on ${\mathcal{C}}^G$ and the resulting action of the representation ring $R_{\mathbb{C}}(G)$ on $H{H}_{\bullet }({\mathcal{C}}^G)$.