2-subnormal not implies automorph-permutable

Property-theoretic statement
The subgroup property of being a 2-subnormal subgroup is not stronger than the subgroup property of being an automorph-permutable subgroup.

Verbal statement
A 2-subnormal subgroup (a subgroup that is normal in a normal subgroup) need not be automorph-permutable.

Related facts

 * 2-subnormal implies conjugate-permutable
 * Automorph-permutable of normal implies conjugate-permutable
 * Automorph-permutable not implies permutable

Corollaries

 * 2-subnormal not implies permutable
 * Automorph-permutability is not transitive
 * Permutability is not transitive: Since normal implies permutable, we see that if permutability were transitive, then any 2-subnormal subgroup would be permutable, and in particular, automorph-permutable.
 * Conjugate-permutable not implies automorph-permutable: This follows because 2-subnormal implies conjugate-permutable.

Example of a dihedral group of order eight
Consider the dihedral group of order eight, generated by a rotation $$a$$ and a reflection $$x$$. Then, the two-element subgroup generated by $$x$$ is a 2-subnormal subgroup (it is not normal, but its normal closure is $$\{ e, a^2, x, a^2x \}$$, which is normal).

On the other hand, the subgroup $$\{ x, e \}$$ does not permute with the subgroup $$\{ ax , e \}$$, which is its image under a suitable outer automorphism.