Left-transitively 2-subnormal implies normal of characteristic

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., left-transitively 2-subnormal subgroup) must also satisfy the second subgroup property (i.e., normal subgroup of characteristic subgroup)
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Statement

Statement with symbols

Suppose $H$ is a subgroup of a group $K$ such that whenever $K$ is a 2-subnormal subgroup of a group $G$, so is $H$. Then, $H$ is a normal subgroup of characteristic subgroup of $K$; in other words, $H$ is a normal subgroup of a characteristic subgroup of $K$.

Related facts

• Characteristic of normal implies normal
• Left transiter of normal is characteristic
• Left residual of 2-subnormal by normal is normal of characteristic
• Characteristic implies left-transitively 2-subnormal

Facts used

1. Left residual of 2-subnormal by normal is normal of characteristic

Proof

Given: $H$ is a subgroup of $K$ such that whenever $K$ is a 2-subnormal subgroup of a group $G$, so is $H$.

To prove: $H$ is a normal subgroup of a characteristic subgroup of $K$.

Proof: Fact (1) states that if $H$ is a subgroup of $K$ such that whenever $K$ is a normal subgroup of some bigger group $G$, $H$ is 2-subnormal in $G$, then $H$ must be a normal subgroup of a characteristic subgroup of $K$.

What we're given is that whenever $K$ is 2-subnormal in $G$ so is $H$. Since normal subgroups are 2-subnormal, fact (1) applies to yield that $H$ is a normal subgroup of a characteristic subgroup of $K$.