Hochschild Cohomology of skew group rings and invariants

Let $A$ be a $k$-algebra and $G$ be a group acting on $A$. We show that $G$ also acts on the Hochschild cohomology algebra $HH^{\bullet}(A)$ and that there is a monomorphism of rings $HH^{\bullet}(A)^G \hookrightarrow HH^{\bullet}(A[G])$. That allows us to show the existence of a monomorphism from $HH^{\bullet}(\widetilde{A})^G$ into $HH^{\bullet}(A)$, where $\widetilde{A}$ is a Galois covering with group $G$.