In this paper, we determine the homotopy groups $\pi_4(\Sigma K(G,1))$,
$\pi_5(\Sigma K(G,1))$ and $\pi_5(\Sigma^2K(G,1))$ for different groups $G$ by
using different facts and methods from group theory and homotopy theory:
derived functors, the Carlsson simplicial construction, the Baues-Goerss
spectral sequence, homotopy decompositions and the methods of algebraic
K-theory.