  Cencelj and Dranishnikov showed that for certain nilpotent groups $G$,
$K(G_{ab},1) \in \text{AE}(X)$ is equivalent to $K(G,1) \in \text{AE}(X)$ for
any compacta $X$ (here $G_{ab}$ is the abelianization of $G$). We examine the
same problem for solvable groups. We also give an elementary proof of this fact
for any nilpotent group and any 2-dimensional metric space.
