Equivariant isotopy of unknots to round circles

Suppose that γ0 is an unknotted simple closed curve contained in the 3-sphere which happens to be invariant under a subgroup G of the Möbius group of S3 = the group (generated by inversions in 2-spheres). It is shown that there is an equivariant isotopy γt, 0 ⩽ t ⩽ 1, from γ0 to a round circle γ1.