We study the isometry groups of compact spherical orientable 3-orbifolds S^3/G, where G is a finite subgroup of SO(4), by determining their isomorphism type. Moreover, we prove that the inclusion of Isom (S^3/G) into Diff (S^3/G) induces an isomorphism of the π0 groups, thus proving the π0-part of the natural generalization of the Smale Conjecture to spherical 3-orbifolds.
