  We prove a decomposition of the Hochschild homology groups of the equivariant
dg category $\mathscr{C}^G$ associated to a small dg category $\mathscr{C}$
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 $\mathscr{C}^G$ and the resulting action of the
representation ring $R_\mathbb{C}(G)$ on $HH_\bullet(\mathscr{C}^G)$.
