Let $G$ be a finite group with finite-dimensional irreducible representations $\rho_i:G\to\mathrm{GL}_{n_i}(k)$ over a field $k$ indexed by $i=1,...,m$. These compose with the canonical map $\mathrm{GL}_{n_i}(k)\to\mathrm{GL}(k)$ to give $\rho_i:G\to\mathrm{GL}(k)$ for $i=1,\cdots,m$. Taking classifying spaces and applying the Quillen plus construction therefore gives maps $\rho_i:BG^+\to B\mathrm{GL}(k)^+$, and hence homomorphisms $\rho_i^n:\pi_n BG^+\to K_n(k)$ for every $1,\cdots,m$, where $K_n(k)$ denotes the $n$th algebraic K-theory group of $k$. Have these maps been studied somewhere? When are these maps surjective?

An easy example is that $\rho_i^{2n}=0$ for $n>0$ if $k=\mathbf{F}_p$. In general, though, I believe studying these maps might be hard, because every path connected space arises as $BG^+$ for some discrete group $G$ (Kan-Thurston). As a special case, what can one say about these maps if $\mathrm{char}(k)=p>0$ and $G$ is a $p$-group?