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Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...

Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...