Base of a wreath product is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., base of a wreath product) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Suppose $H$ is the base of a wreath product in $K$ and $K$ is the base of a wreath product in $G$. Then, $H$ is the base of a wreath product in $G$.

Related facts

• Base of a wreath product implies right-transitively 2-subnormal
• Base of a wreath product implies right-transitively conjugate-permutable

Facts used

1. Wreath product is associative