This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb F_q$. By Quillen, $\operatorname K_3(\mathbb F_q)$ is $\mathbb Z/(q^2-1)$. Since $K_2(\mathbb F_q)$ is zero, by the universal coefficient theorem for any abelian group $A$ and sufficiently large $n$ the group $H^3(\operatorname{SL}_n(\mathbb F_q);A)$ is isomorphic to the $q^2-1$-torsion subgroup of $A$. More precisely, there is a universal class in $H^3(\operatorname{SL}_n(\mathbb F_q);\mathbb Z/(q^2-1))$ such that any class for any $A$ is induced by a unique homomorphism $\mathbb Z/(q^2-1)\to A$.
On the other hand, for any finite normal extension of global fields $K/k$, to a finite-dimensional simple $K$-algebra $S$ with the property that any $k$-automorphism of $K$ extends to an automorphism of $S$, corresponds the Teichmüller class $\xi_S\in H^3(\operatorname{Gal}(K/k);K^*)$ representing the obstruction for $S$ to be extended from a simple $k$-algebra. The latter cohomology is actually cyclic, with the generator being the Teichmüller class of an algebra.
Can these two fit together somehow? That is, given a ($q^2-1$)st root of unity in $K$, can one explicitly construct a Galois representation over $\mathbb F_q$ which would relate the universal class in $H^3(\operatorname{SL}_n(\mathbb F_q);\mathbb Z/(q^2-1))$ to the Teichmüller class in $H^3(\operatorname{Gal}(K/k);K^*)$ through the homomorphisms $\operatorname{Gal}(K/k)\to\operatorname{SL}_n(\mathbb F_q)$ and $\mathbb Z/(q^2-1)\to K^*$?
