Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of $\mathbb{C}$-valued sheaves).

They also mention that it should be possible to define $\displaystyle D^b_{G(\overline{\mathbb{F}_p})}(X(\overline{\mathbb{F}_p}), \overline{\mathbb{Q}_l})$

So my question: Do the comparison theorems of [BBD] Chapter 6 also hold in the equivariant case?

Is there any "nice" reference?

[BL94] Equivariant Sheaves and Functors, Bernstein, J.; Lunts. V. Springer Lecture Notes. [BBD] Beilinson, A. A.; J. Bernstein, P. Deligne (1982). "Faisceaux pervers". Astérisque (Société Mathématique de France, Paris) 100.