Conjugate generation of sporadic almost simple groups

As defined by Guralnick and Saxl, given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number ${\alpha }_S(x)$ is the minimum number of conjugates of $x$ in $⟨x,S⟩$ that generate a subgroup containing $S$. In this paper, for every sporadic group $S$ other than the Monster and an automorphism $x$ of $S$ of prime order, we complete the determination of the precise value of ${\alpha }_S(x)$.