How can I detect the homology image of a unipotent group in the general linear group?

Question

Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements.

Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent group in the sense that it consists of $n\times n$ upper-triangular matrices with 1's on the diagonal.

The inclusion $U_n(F)\hookrightarrow\text{GL}_n(F)$ induces a (unstable) homology homomorphism $f_k: H_k(U_n(F);\mathbb{Z})\to H_k(\text{GL}_n(F);\mathbb{Z})$ for any positive integer $k$.

And there is another map, a stable homology map: $g_k: H_k(U_n(F);\mathbb{Z})\to H_k(\text{GL}(F);\mathbb{Z})$.

How can I detect the image of $f_k$ (or $g_k$)? Is $\text{im}(f_k)$ or $\text{im}(g_k)$ always zero?

Background: In Suslin's paper [1] (Sublemma 4.4.2), he asserts that when $k=n=\text{char}(F)=3$ then $\text{im}(f_k)=\text{im}(g_k)=0$ with a reference directed to [2]. But from [2] it's quite unclear to me how to deduce the assertion made in [1].

[1] A. Suslin, K3 of a field, and the Bloch group (Russian), Trudy Mat. Inst. Steklov 183 (1990), 180-199. English transl. in Proc. Steklov Inst. Math. (1991), 217-239.

[2] A. Suslin, Stability in algebraic K-theory, pp. 304-333 in Lecture Notes in Math. 966, Springer-Verlag, 1982.

Answer 1

Suppose first that $F$ is a finite field of characteristic $p$. Then $U_n(F)$ is a Sylow $p$-subgroup of $GL_n(F)$, and so using the transfer in group homology one sees that the image of $f_k$ (for $k>0$) is precisely the $p$-torsion in $H_k(GL_n(F);\mathbb{Z})$. Unstably, this is not well understood, but it cannot be trivial by general principles in group cohomology (see the top answer to this MO question). But it follows that the image of $g_k$ is contain in the $p$-torsion of $H_k(GL(F);\mathbb{Z})$, and this is completely understood by Quillen's calculations: it is trivial.

Suppose now that the field $F$ is infinite; it is then a "ring with many units" in the sense of Nesterenko--Suslin (Homology of the full linear group over a local ring, and Milnor's K-theory), and we may apply the results of that paper. There is a group $UT_n(F)$ of all upper triangular matrices (not necessarily having 1's on the diagonal), and a split extension $$1 \to U_n(F) \overset{i}\to UT_n(F) \overset{q}\to D_n(F) \to 1$$ where $D_n(F)$ are the diagonal matrices. If follows by iteratedly applying Theorem 1.11 of Nesterenko--Suslin that the map $q$ induces an isomorphism in integral homology, but then $i$ induces the trivial map in integral homology. As the inclusion of $U_n(F)$ into $GL_n(F)$ factors over $UT_n(F)$, it follows that $f_k$ (and hence $g_k$) is trivial.