Say I have a smooth scheme X over a filed with an action of a finite group G such that the quotient X/G exists in the category of smooth schemes. I would like to know how this quotient is related to the (homotopy) quotient taken in the category of motives. The map X-->X/G is an étale cover so the presheaf represented by X/G and the quotient by G of the presheaf represented by X become equivalent after etale sheafification.

However I don't see why they should be equivalent if we only sheafify for the Nisnevich topology.