Diagonal subgroup of a wreath product

Definition
A subgroup $$G$$ of a group $$L$$ is termed a diagonal subgroup of a wreath product if we can express $$L$$ as an internal wreath product (the internal version of external wreath product) with $$G$$ as the diagonal subgroup corresponding to the base direct power.

Related properties

 * Base of a wreath product: Note that for any wreath product, the base and diagonal are isomorphic as abstract groups. However, they need not be automorphic subgroups. In fact, while the base is always a 2-subnormal subgroup, the diagonal need not be subnormal at all, and in fact it is subnormal if and only if (as an abstract group) it is a nilpotent group (note that the whole group still need not be nilpotent).