2-subnormal implies conjugate-permutable

Verbal statement
Any 2-subnormal subgroup (i.e. a subgroup that is normal inside its normal closure) is conjugate-permutable.

Symbolic statement
Suppose $$H$$ is a 2-subnormal subgroup of $$G$$. In other words, $$H$$ is a normal subgroup inside its normal closure in $$G$$. Then, for any $$g \in G$$, $$H$$ and $$gHg^{-1}$$ are permuting subgroups.

Property-theoretic statement
The subgroup property of being 2-subnormal is stronger than the subgroup property of being conjugate-permutable.

Facts used

 * Automorph-permutable of normal implies conjugate-permutable

Proof modulo the fact on automorph-permutable subgroups
A 2-subnormal subgroup is an normal subgroup of a normal subgroup. Since any normal subgroup is automorph-permutable, a 2-subnormal subgroup is an automorph-permutable subgroup of a normal subgroup. Using the above fact, we conclude that any 2-subnormal subgroup is conjugate-permutable.