Generation by conjugate elements of finite almost simple groups with a sporadic socle

As defined by Guralnick and Saxl given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number ${\alpha }_S(x)$ does not exceed a natural number $m$ if some $m$ conjugates of $x$ in the group $⟨x,S⟩$ generate a subgroup that includes $S$. The outcome of this paper together with one by Di Martino, Pellegrini, and Zalesski, both of which are based on computer calculations with character tables, is a refinement of the estimates by Guralnick and Saxl on the value of ${\alpha }_S(x)$ in the case where $S$ is a sporadic group. In particular, we prove that ${\alpha }_S(x)⩽4$, except when $S$ is one of the Fischer groups and $x$ is a $3$-transposition. In the latter case, ${\alpha }_S(x)=6$ if $S$ is either $F{i}_{22}$ or $F{i}_{23}$ and ${\alpha }_S(x)=5$ if $S={F{i}_{24}}^{\mathrm{\prime }}$.