Isometry groups and mapping class groups of spherical 3-orbifolds

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3/G$, where $G$ is a finite subgroup of $\mathrm{SO}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $\mbox{Isom}(S^3/G)$ into $\mbox{Diff}(S^3/G)$ 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.