2-subnormality is not transitive

Statement
A 2-subnormal subgroup of a 2-subnormal subgroup need not be 2-subnormal.

Failures of transitivity

 * Normality is not transitive
 * There exist subgroups of arbitrarily large subnormal depth
 * Descendant not implies subnormal
 * Ascendant not implies subnormal

Left-transitively 2-subnormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed left-transitively 2-subnormal in $$G$$ if whenever $$G$$ is 2-subnormal in some group $$K$$, so is $$H$$. Since any characteristic subgroup of a normal subgroup is normal, every characteristic subgroup is left-transitively 2-subnormal.

Right-transitively 2-subnormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed right-transitively 2-subnormal in $$G$$ if whenever $$K$$ is a 2-subnormal subgroup of $$H$$, $$K$$ is 2-subnormal in $$G$$. Any transitively normal subgroup, as well as any base of a wreath product, is right-transitively 2-subnormal.

Facts used

 * 1) uses::There exist subgroups of arbitrarily large subnormal depth

Proof
The proof follows directly from fact (1): if every 2-subnormal subgroup of a 2-subnormal subgroup were 2-subnormal, then every subnormal subgroup would be 2-subnormal, and we would not get subgroups of arbitrarily large subnormal depth.