This question is inspired by Terry Tao's blog post and the comments there. I have always cited M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. Szeged 14, pp. 69-82 (1951) as the source of the result that if $G$ is a group and $H$ is a subgroup, then the regular action of $G$ on itself is inside of the action of the wreath product $H\wr (G/\mathrm{Core}(H))$ on $H\times G/H$ via the identification of $G$ with $H\times G/H$ using a transversal $T$ for $G/H$ and that replacing $T$ by a different transversal changes the embedding of $G$ by an inner automorphism of the wreath product.
I am aware that this result is implicit in Frobenius's theory of induced representations and the theory of monomial representations of groups (e.g. Chapter 4 this paper of Ore although $H$ is assumed finite index there) but these authors did not use the wreath product language to the best of my knowledge.
Many people refer only to the special case where $N$ is normal as the Krasner-Kaloujnine embedding. I am now wondering which version they actually stated. The original paper does not seem easy to access by the obvious Google search.