I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question.

Let G be a topological group and X be a G-space (for a nice notion of "space"). He defines

$${X}_{hG}=EG{\times}_{G}X$$

the homotopy orbit space, and

$${X}^{hG}=F(EG,X{)}^{G}$$

the homotopy fixed point space.

He claims there are spectral sequences

$${E}_{p,q}^{2}={H}_{p}(G;{H}_{q}(X))\Rightarrow {H}_{p+q}({X}_{hG})$$

and

$${E}_{2}^{p,q}={H}^{-p}(G;{\pi}_{q}(X))\Rightarrow {\pi}_{p+q}({X}^{hG}).$$

Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration $X\to {X}_{hG}\to BG$ (take the fiber bundle with fiber X associated to the action of G on X and to the G-principal bundle $EG\to BG$).

But where does the second one come from?

Let G be a topological group and X be a G-space (for a nice notion of "space"). He defines

$${X}_{hG}=EG{\times}_{G}X$$

the homotopy orbit space, and

$${X}^{hG}=F(EG,X{)}^{G}$$

the homotopy fixed point space.

He claims there are spectral sequences

$${E}_{p,q}^{2}={H}_{p}(G;{H}_{q}(X))\Rightarrow {H}_{p+q}({X}_{hG})$$

and

$${E}_{2}^{p,q}={H}^{-p}(G;{\pi}_{q}(X))\Rightarrow {\pi}_{p+q}({X}^{hG}).$$

Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration $X\to {X}_{hG}\to BG$ (take the fiber bundle with fiber X associated to the action of G on X and to the G-principal bundle $EG\to BG$).

But where does the second one come from?