A note on the Kaloujnine-Krasner theorem

The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence (Formula presented.) of groups and a section (Formula presented.), an embedding (Formula presented.) 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 (Formula presented.). Moreover, one says that two extensions (Formula presented.) and (Formula presented.) of H by N are equivalent if there exists a group isomorphism (Formula presented.) such that (Formula presented.) and (Formula presented.). We say that two embeddings (Formula presented.) and (Formula presented.) are equivalent if there exists a group isomorphism (Formula presented.) such that (Formula presented.). We show that two extensions (Formula presented.) and (Formula presented.) are equivalent if and only if the embeddings (Formula presented.) and (Formula presented.), associated with any two sections (Formula presented.) and (Formula presented.) via the Kaloujnine-Krasner theorem, are equivalent.