Isometry groups and mapping class groups of spherical 3-orbifolds

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3\mathrm{/}G$, where $G$ is a finite subgroup of $\mathrm{S}\mathrm{O}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $Isom(S^3\mathrm{/}G)$ into $Diff(S^3\mathrm{/}G)$ induces an isomorphism of the ${\pi }_0$ groups, thus proving the ${\pi }_0$-part of the natural generalization of the Smale Conjecture to spherical $3$-orbifolds.