A normalized holomorphic motion of a closed set in the Riemann
sphere, defined over a simply connected complex Banach manifold,
can be extended to a normalized quasiconformal motion of the
sphere, in the sense of Sullivan and Thurston. In this paper,
we show that if the given holomorphic motion, defined over
a simply connected complex Banach manifold, has a group equivariance
property, then the extended (normalized) quasiconformal motion
will have the same property. We then deduce a generalization
of a theorem of Bers on holomorphic families of isomorphisms
of Möbius groups. We also obtain some new results on
extensions of holomorphic motions. The intimate relationship
between holomorphic motions and Teichmüller spaces
is exploited throughout the paper.