Is there an example of a nontrivial discrete amenable group with vanishing integral homology?

To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the Johnson cocycle in $Z^1_b(Q, (\ell^{\infty}(Q)/\mathbb R)^*)$ (universal cocycle obstructing amenability, which sends pair of elements $(g_1, g_2)$ to the difference of delta measures supported on them) to be nontrivial?

It is well known that every perfect group is a homomorphic image of some acyclic group. If the answer to the question is negative, can we say something about homology of the kernels of those acyclic covers of perfect amenable groups?