# Do homotopy pullbacks commute with homotopy orbits (in spaces)?

Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h_Z Y)_{hG}$ weakly equivalent to $X_{hG} \times^h_{Z_{hG}} Y_{hG}$?

Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration.
$\to \pi_* (U) \to \pi_*(U_{hG}) \to \pi_*(BG) \to \dots$