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 $\langle x,S\rangle$ 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 $\alpha_S(x)\leqslant 4$, 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 $Fi_{22}$ or $Fi_{23}$ and $\alpha_S(x)=5$ if $S={Fi_{24}}'$.