  Let $G$ be a group. Let $\mathrm{Perm}(G)$ denote its symmetric group and
write $\mathrm{Hol}(G)$ for the normalizer of the subgroup of left translations
in $\mathrm{Perm}(G)$. The multiple holomorph $\mathrm{NHol}(G)$ of $G$ is in
turn defined to be the normalizer of $\mathrm{Hol}(G)$ in $\mathrm{Perm}(G)$.
In this paper, we shall show that the quotient group
$\mathrm{NHol}(G)/\mathrm{Hol}(G)$ has order two when $G$ is finite and almost
simple. As an application of our techniques, we shall also develop a method to
count the number of Hopf-Galois structures of isomorphic type on a finite
almost simple extension in terms of fixed point free endomorphisms.
