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This is not an answer but an attempt to clarify the question. In the category of right $G$-spaces (with weak equivalences being the maps that as maps of spaces are weak equivalences) let us single ou …

I thought about the case when $G$ is a free group. I believe that in this case $H_2(BG/G)$ is the second exterior power of $G^{ab}$ (which is also $H_2(BG^{ab})$), and $H_3(BG/G)$ is the kernel of $$ …

Of course, in any spectral sequence $E_{r+1}$ is a subquotient of $E_r$ (the kernel of $d_r$ divided by the image of $d_r$). And in general new torsion can appear in the sense of torsion elements in $ …

If I understand the question right, I think the classifying space can be described like this. Let $Map(G,BH)$ be the space of all continuous maps from $G$ to $BH$. Make the group $G$ act continuously …

I think that this works. EDIT: No, it doesn't. See John Rognes's comment. Notation: For a point $x\in EG$ we may symbolically write $x=\sum_{j=0}^nt_jg_j$, where $g_0,\dots ,g_n$ is an ordered tuple o …

A lifting of the map $Fr(M)\to EO(n)$ along the map $(B\phi^\ast)EO(n)\to EO(n)$ always exists (up to homotopy, which I think is what you mean), simply because the space $EO(n)$ is contractible and th …