  An extremal quasiconformal homeomorphisms in a class of homeomorphisms
between two CR 3-manifolds is an one which has the least conformal distortion
among this class. This paper studies extremal quasiconformal homeomorphisms
between CR 3-manifolds which admit transversal CR circle actions. Equivariant
$K$-quasiconformal homeomorphisms are characterized by an area-preserving
property and the $K$-quasiconformality of their quotient maps on the spaces of
$S^1$-orbits. A large family of invariant CR structures on $S^3$ is constructed
so that the extremal quasiconformal homeomorphisms among the equivariant
mappings between them and the standard structure are completely determined.
These homeomorphisms also serve as examples showing that the extremal
quasiconformal homeomorphisms between two invariant CR manifolds are not
necessarily equivariant.
