Here is a hopelessly naive question. Please point me to the relevant literature!

Let $F$ be any field. The Cartan subgroup of $\mathrm{PGL}_2(F)$ is $F^\times\rtimes \Sigma_2$. Let $X_2(F)$ be the homotopy cofiber of $B(F^\times\rtimes\Sigma_2)\to B\mathrm{PGL}_2(F)$, so there is a long exact sequence
$$
\ldots\to H_i(F^\times\rtimes\Sigma_2)\to H_i(\mathrm{PGL}_2(F))\to H_i(X_2(F))\to \ldots
$$
(I'm not sure if I should write $H_i(G)$ or $H_i(BG)$...).
 If I am not making a stupid mistake, then the known computations of the homology of $\mathrm{GL}_2(F)$ going up to the first unstable group $H_3(\mathrm{GL}_2(F))$ can be summarized by saying that $H_i(X_2(F))=0$ for $i=1,2$, and $H_3(X_2(F))=\mathfrak p(F)$ is the pre-Bloch group of $F$. (The pre-Bloch group is the quotient of $\mathbb Z[F^\times]$ by the ``$5$-term relations''.) (I might be off by some $2$-torsion.) By Hurewicz (and the check that $\pi_1 X_2(F)=0$), the same holds true for the homotopy groups of $X_2(F)$.

As far as I could find, little is known about the homology of $\mathrm{GL}_2(F)$ or $\mathrm{PGL}_2(F)$ in degrees $>3$. The rational structure could be determined by the following very naive question.

> Question. Are the homotopy groups $\pi_i X_2(F)$ bounded torsion for $i\neq 3$?

I might even expect the implicit bound to be independent of $F$ (but of course depend on $i$; the order of magnitude should be $i!$).

The only evidence I have is that a back-of-the-envelope calculation seems to suggest that this holds true for finite fields. In that case the order of $\mathrm{PGL}_2(\mathbb F_q)$ is the product of $q-1$, $q$ and $q+1$. Localized at $q-1$, the homology agrees with the homology of the Cartan; localized at $q$, it is only in degrees larger than $q$ (which is OK as the bound on torsion may depend on $i$), and localized at $q+1$, the homology seems to agree with the homology of a $K(\mathbb Z/(q+1),3)$ (up to bounded torsion in each degree), which is roughly $\mathbb Z/(q+1)$ in all degrees $\equiv 3\mod 4$ (and is bounded torsion in other degrees). The pre-Bloch group $\mathfrak p(\mathbb F_q)$ is also $\mathbb Z/(q+1)$ up to $2$-torsion, so things add up.

The case of $F=\mathbb Q$ might be amenable to computations, but I was unable to do those. Are there any relevant results about $H_4(\mathrm{GL}_2(F))$ that might shed light on $\pi_4 X_2(F)$?

A final remark: I expect that it is critical that $F$ is a field. The similar statement should be false already for discrete valuation rings, or $\mathbb Z[\tfrac 1n]$ (which is why the case of $F=\mathbb Q$ is not so easy to compute; I presume one would try to first compute for all $\mathbb Z[\tfrac 1n]$ and then pass to a colimit, and the desired structure should only appear in the colimit).

**Update:** As mentioned by Matthias Wendt in the comments, a special case of the work of Borel-Yang says that $H_i(\mathrm{PGL}_2(\mathbb Q),\mathbb Q)=0$ for $i>0$. Inserting this into the above long exact sequence, one immediately sees that $X_2(\mathbb Q)$ is in fact rather complicated, and for example has nontorsion $\pi_5$. Thus, the answer to the question is a strong No. Thanks for the help in sorting this out!