  We prove in this paper that a finite almost simple group $R$ with socle the
non-abelian simple group $S$ possesses a conjugacy class of core-free maximal
subgroups whose index coincides with the smallest index $\operatorname{l}(S)$
of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups
with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$.
We show that the number of subgroups of the outer automorphism group of $S$ is
bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.
