  We link small modifications of projective varieties with a ${\mathbb
C}^*$-action to their GIT quotients. Namely, using flips with centers in
closures of Bia{\l}ynicki-Birula cells, we produce a system of birational
equivariant modifications of the original variety, which includes those on
which a quotient map extends from a set of semistable points to a regular
morphism.
  The structure of the modifications is completely described for the blowup
along the sink and the source of smooth varieties with Picard number one with a
${\mathbb C}^*$-action which has no finite isotropy for any point. Examples can
be constructed upon homogeneous varieties with a ${\mathbb C}^*$-action
associated to short grading of their Lie algebras.
