Quotient of a motive by a finite group

Question

Given a smooth scheme $X$ over a field $k$, we can consider its motive $M(X)$. It is an object in Voevodsky's triangulated category of motives $DM(k,\Lambda)$ where $\Lambda$ is the ring of coefficients. It is constructed from the presheaf with transfers represented by $X$ by forcing $A^1$-invariances and invertibility of the Lefschetz motive. Now, if $X$ is acted on by a finite group in such a way that the categorical quotient $X/G$ exists in the category of smooth schemes, I can consider the motive $M(X/G)$. On the other hand, without any assumption on the action, I can construct a motive $M(X)_G$ by taking the homotopy quotient of $M(X)$ by $G$ in $DM(k,\Lambda)$. There is a slight abuse here as one needs to work at the point set level (using model categories or infinity-categories) in order to define this homotopy quotient. As far as I understand $M(X)_G$ is also the motive of the quotient stack $[X/G]$ Clearly there is a map $$M(X)_G\to M(X/G)$$ and I am wondering whether this map is an isomorphism. I think that it is the case when we work in $DM_{et}(k,\Lambda)$, the variant of $DM(k,\Lambda)$ that uses the étale topology instead of the Nisnevich topology because in that case the map $X\to X/G$ is an étale cover. Is it also true in $DM(k,\Lambda)$ ? And if it is not true in general are there conditions under which it is known to be true ?