Let $\mathcal C$ be a permutative category, that is a symmetrical monoidal category with strict associativity. One can then define the $K$-groups of $\mathcal C$, for $n >0$ by $$K_n(\mathcal C) = \pi_n(\Omega B |\mathcal C|),$$ where $|C|$ denotes the realization of the nerve of $\mathcal C$ that inherits a multiplication coming from the monoidal structure.

My question is: If all Hom-sets in $\mathcal C$ are finite, are the groups $K_n(\mathcal C)$ then also all finite?

Here are there three examples I have seen that motivated this question:

(1) If $\mathcal C$ is given by an abelian group $G$, we get $\Omega B |\mathcal C| = BG$.

(2) If $\mathcal C$ is the category of finite sets with inclusions, we have $\Omega B |\mathcal C| = (B\Sigma_{\infty})^+ = Q_0S^0$ by Baratt-Priddy-Quillen, hence we get the stable homotopy groups of spheres.

(3) For $R$ a (commutative) ring, we can take $\mathcal C$ to be free modules of finite rank and get the higher algebraic $K$-groups of $R$. Quillen computed them for $R = \mathbb F_q$ a finite field, they are $K_{2i} = 0$ and $K_{2i-1} = \mathbb Z/(q^i-1)$.

Disclaimer: I have just started to learn about higher K-theory and this question is motivated by my ignorance, so feel free to close it if it's stupid.