Subnormal automorphism-invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subnormal automorphism-invariant subgroup if, given any automorphism $$\varphi$$ of $$G$$ such that $$\varphi$$ restricts to an automorphism for every subnormal subgroup of $$G$$, $$\varphi(H) = H$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Subnormal subgroup
 * Weaker than::Join of finitely many subnormal subgroups
 * Weaker than::Join of subnormal subgroups
 * Weaker than::Intersection of subnormal subgroups