Let G be a group. Let Perm(G) denote its symmetric group and write Hol(G) for the normalizer of the subgroup of left translations in Perm(G). The multiple holomorph NHol(G) of G is defined to be the normalizer of Hol(G) in Perm(G). In this paper, we shall show that the quotient group NHol(G)/Hol(G) has order two whenever 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 field extension in terms of fixed point free endomorphisms.

The author would like to thank the referee for helpful comments.