Automorph-permutable not implies permutable

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., automorph-permutable subgroup) need not satisfy the second subgroup property (i.e., permutable subgroup)
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Statement

Verbal statement

An automorph-permutable subgroup of a group need not be permutable.

Facts used

• Symmetric groups on finite sets are complete: For $n\ne 2,6$, the symmetric group on $n$ letters is complete: it is centerless and every automorphism of it is inner.

Proof

Example in the symmetric group on four letters

Consider $G$ to be the symmetric group on four letters: $\{1,2,3,4\}$. Consider the two-element subgroup $H$ generated by the double transposition $(12)(34)$.

$H$ is an automorph-permutable subgroup in $G$: Since symmetric groups are complete, it suffices to argue that $H$ is conjugate-permutable in $G$. This, in turn follows because $H$ is a 2-subnormal subgroup of $G$: it is normal in the subgroup $K=\{(),(13)(24),(12)(34),(14)(23)\}$, which is normal in $G$.

On the other hand, $H$ is not permutable in $G$, for instance, $H$ does not permute with the subgroup generated by $(123)$.