Base of a wreath product is transitive

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) uses::Wreath product is associative