Cofactorial automorphism-invariant implies left-transitively 2-subnormal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., cofactorial automorphism-invariant subgroup) must also satisfy the second subgroup property (i.e., left-transitively 2-subnormal subgroup)
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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Cofactorial automorphism-invariant subgroup (?) and 2-subnormal subgroup (?)), to another known subgroup property (i.e., 2-subnormal subgroup (?))
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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

Facts used

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

Proof

The proof follows from Fact (1).