Base of a wreath product with diagonal action

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Definition

A subgroup $G$ of a group $L$ is termed a base of a wreath product with diagonal action if $L$ can be expressed as an internal wreath product with diagonal action with $G$ as base. In other words, $L$ is an internal semidirect product of a direct power $H$ of $G$ (with $G$ as one of the factors) and a subgroup $K_{1}\times K_{2}$ where $K_{1}$ acts by coordinate permutations and $K_{2}$ acts diagonally by automorphisms on each coordinate.