In more classical terms, a superperfect group is one whose abelianization and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.
For example, if G is the fundamental group of a homology sphere, then G is superperfect.
The alternating group A5 is perfect but not superperfect: it has a non-trivial central extension, the binary icosahedral group (which is in fact its UCE) is superperfect.
More generally, the projective special linear groups PSL(n, q) are simple (hence perfect) except for PSL(2, 2) and PSL(2, 3), but not superperfect, with the special linear groups SL(n,q) as central extensions.
Every acyclic group is superperfect, but the converse is not true: the binary icosahedral group is superperfect, but not acyclic.