Suppose $C$ is a $k$-linear abelian category with an action of a linear algebraic group $G/k$. Suppose $C$ has enough projectives/injectives so I can form the bounded derived category $D(C)$. Under what conditions will it be true that $D(C^G)$, the category of $G$-equivariant objects in $C$ [suppose this still has enough projectives/injectives], maps equivalently to $D(C)^G$, the $G$-equivariant derived category?
Example: $C$ is a category of coherent sheaves on a variety $X$ with a $G$-action.
