We prove here that with a very small number of exceptions, when $G$ is a sporadic simple group and $p$ is a prime such that the Sylow $p$-subgroups of $G$ are nonabelian, then $Out(G)$ is isomorphic to the outer automorphism groups of the fusion and linking systems of $G$. In particular, the $p$-fusion system of $G$ is tame in the sense of [AOV1], and is tamely realized by $G$ itself except when $G\cong M_{11}$ and $p=2$. From the point of view of homotopy theory, these results also imply that $Out(G)\cong Out(BG^\wedge_p)$ in many (but not all) cases.