  Let G be a finite group. It has recently been proved that every nontrivial
element of G is contained in a generating set of minimal size if and only if
all proper quotients of G require fewer generators than G. It is natural to ask
which finite groups, in addition, have the property that any two elements of G
that do not generate a cyclic group can be extended to a generating set of
minimal size. This note answers the question. The only such finite groups are
very specific affine groups: elementary abelian groups extended by a cyclic
group acting as scalars.
