Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $X' \to X$ be a finite etale cover. How Grothendieck groups $K_0(X)$ and $K_0(X')$ are related in this situation?
Set $G$ to be the group of deck transformations of the cover, is it true that $K_0(X) \cong K_0(X') \otimes_{\mathbb{Z}} K_0(\mathbb{C}[G])$?