Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal

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