Normal of characteristic of base of a wreath product with diagonal action implies 2-subnormal

Statement
Suppose $$H \le K \le L \le G$$ are groups such that $$L$$ is a base of a wreath product with diagonal action in $$G$$, $$K$$ is a characteristic subgroup of $$L$$, and $$H$$ is a normal subgroup of $$K$$, then $$H$$ is a 2-subnormal subgroup of $$G$$.

Related facts

 * Base of a wreath product is transitive (this follows from the fact that wreath product is associative)
 * Base of a wreath product implies right-transitively 2-subnormal