  Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial)
factorization of $G$ is an expression of the form $G=HK$, where the factors $H$
and $K$ are core-free subgroups. There is an extensive literature on
factorizations of almost simple groups, with important applications in
permutation group theory and algebraic graph theory. In a recent paper, Li and
Xia describe the factorizations of almost simple groups with a solvable factor
$H$. Several infinite families arise in the context of classical groups and in
each case a solvable subgroup of $G_0$ containing $H \cap G_0$ is identified.
Building on this earlier work, in this paper we compute a sharp lower bound on
the order of a solvable factor of every almost simple group and we determine
the exact factorizations with a solvable factor. As an application, we describe
the finite primitive permutation groups with a nilpotent regular subgroup,
extending classical results of Burnside and Schur on cyclic regular subgroups,
and more recent work of Li in the abelian case.
