A note on the Kaloujnine-Krasner theorem

The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence 1 -> N -> (iota) G -> (pi) H -> 1 of groups and a section s:H -> G, an embedding Phi : G -> N(sic)H of G into the (unrestricted) wreath product of N and H. Given two groups H and N, a short exact sequence as above is called an extension of H by N, denoted by (G;iota,pi). Moreover, one says that two extensions (G(1);iota(1),pi(1)) and (G(2);iota(2),pi(2)) of H by N are equivalent if there exists a group isomorphism eta : G(1) -> G(2) such that iota(2)=eta circle iota(1) and pi(1)=pi(2)circle eta. We say that two embeddings Phi(1):G(1) -> N(sic)H and Phi(2):G(2)-> N(sic)H are equivalent if there exists a group isomorphism eta : G(1) -> G(2) such that Phi(1)=Phi(2) circle eta. We show that two extensions (G(1);iota(1),pi(1)) and (G(2);iota(2),pi(2)) are equivalent if and only if the embeddings Phi(1) and Phi(2), associated with any two sections s(1 ): H -> G(1) and s(2 ): H -> G(2) via the Kaloujnine-Krasner theorem, are equivalent.