  Consider an extension of groups 1 -> K -> G -> Q -> 1 which enjoys the
property that the quotient by the lower central series Gamma_{c+1} produces
another extension 1 -> K/ Gamma_{c+1} K -> G /Gamma_{c+1} G -> Q / Gamma_{c+1}
Q -> 1, of nilpotent groups of class c. We say that the extension is
Gamma_{c+1}-flat. Let us pull back the original extension along any
homomorphism X -> Q. Does the pullback extension enjoy the same
Gamma_{c+1}-flatness property?
  To answer this question we consider not only quotients by the lower central
series, but any localization functor in the category of groups. In fact we
start by studying the analogous question for spaces, where we replace
extensions by fibration sequences. We prove that the only homotopical
localization functors which behave well under pull-backs are nullifications. In
the category of groups, nullifications also enjoy this property, and so do all
epireflections arising from a variety of groups. In particular the answer to
the question about the nilpotent quotients is positive.
