Cofactorial automorphism-invariant implies left-transitively 2-subnormal

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a cofactorial automorphism-invariant subgroup of $$K$$ and $$K$$ is a 2-subnormal subgroup of $$G$$. Then, $$H$$ is a 2-subnormal subgroup of $$G$$.

Related facts

 * Characteristic of normal implies normal
 * Left transiter of normal is characteristic
 * Normal not implies left-transitively fixed-depth subnormal
 * Normal not implies right-transitively fixed-depth subnormal