Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\to H_3(\text{GL}_4(\mathbb{F}))$?
Is it an isomorphism?(Here $H_n(G)$ means the $n$-th integral homology of the group $G$)

Here is some material I have known. [Sprehn&Wahl](https://arxiv.org/abs/1812.08742v2) says $f$ must be a surjection. In the paper [1], it is shown that $f$ would be an isomorphism if $\mathbb{F}$ is an infinite field. But in the talk [[2]](https://people.math.harvard.edu/~kupers/notes/nrwtalk.pdf), it is asserted that the techniques in [1] doesn't apply for finite fields. And with the result of [Galatius-Kupers-Randal-Williams](https://arxiv.org/abs/1810.11931v1), it is deduced that $f$ induces an isomorphism on $p$-primary part. I'm wondering what happens with the $\text{mod } l$ homology where $l\neq p$.

The reason I raise this question is that in the Chapter.VI in Weibel's [K-book, remark 5.12.1](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.VI.pdf), he asserts the map $\varphi$ could extend to a map $H_3(\text{GL}(\mathbb{F}))\to B(F)$. There one works on a general field $F$ with $|F|>3$. But all I can get now is $ H_3(\text{GL}(\mathbb{F}))=H_3(\text{GL}_4(\mathbb{F})) $ and if we want to extend $\varphi$ to $H_3(\text{GL}(\mathbb{F}))$ we have to work on $\ker(f)$.

reference:

[1]*Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a
local ring, and Milnor’s K-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1,
121–146. MR 992981*

[2]*Alexander Kupers, [NRW TALK: $E_\infty$-CELLS AND THE HOMOLOGY OF GENERAL
LINEAR GROUPS OVER FINITE FIELDS](https://people.math.harvard.edu/~kupers/notes/nrwtalk.pdf)*