Perfect Radicals and Homology of Group Extensions

A group epimorphism Q: G+ Q preserves perfect radicals if $PG = PQ, where PC, PQ is maximal perfect in G, Q respectively. The following two questions are considered. Question 1. If P,,(B) acts trivially on H,(F; E), does the tibration F + Es B then have ~,(P)(P~,(E)) = Pm,(B)? Question 2. If f: X + Y is a map of spaces which induces an isomorphism of homology groups, is rr,(X) * rr,( Y) n,( Y)/Prl( Y) then an epimorphism? It is shown that each question is equivalent to its group-theoretic counterpart obtained when the spaces involved are classifying spaces of discrete groups. It is also shown that an affirmative answer to the second question implies an affirmative answer to Question 1. By means of a direct product construction on finite nilpotent groups a continuum of examples is exhibited to resolve these question in the negative. This leaves to an example of an inclusion map of a locally finite p-group in a countable hypoabelian group which induces an isomorphism of all homology groups.