Automorph-permutable not implies permutable

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.

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 $$(123)$$.