Permutability is not transitive

Property-theoretic statement
The subgroup property of being a permutable subgroup does not satisfy the subgroup metaproperty of being a transitive subgroup property.

Verbal statement
A permutable subgroup of a permutable subgroup need not be permutable.

Facts used

 * Normal implies permutable
 * 2-subnormal not implies automorph-permutable
 * Permutable implies automorph-permutable

Proof
We prove this by contradiction, using the fact that a 2-subnormal subgroup need not be automorph-permutable.

Suppose permutability were transitive. Then, since, every normal subgroup is permutable, every 2-subnormal subgroup would be a permutable subgroup of a permutable subgroup, and hence permutable (by transitivity). Since every permutable subgroup is automorph-permutable, it would follow that every 2-subnormal subgroup is automorph-permutable, a contradiction.