Subnormal stability automorphism-invariant subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a subnormal stability automorphism-invariant subgroup if, for any subnormal series of $$G$$, and any stability automorphism $$\sigma$$ of that subnormal series, $$\sigma(H) = H$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Cofactorial automorphism-invariant subgroup for a finite group.

Weaker properties

 * Stronger than::Normal stability automorphism-invariant subgroup

Facts

 * For a finite group, the stability automorphisms of any subnormal series have no other prime factors to their order than the prime factors of the order of the group.
 * For a group of prime power order, the stability automorphisms of subnormal series are precisely the same as the $$p$$-automorphisms.