Auto-invariance property

Definition with symbols
A subgroup property $$p$$ is termed an auto-invariance property if there is a group-closed automorphism property $$q$$ such that for any subgroup $$H \le G$$, $$H$$ has property $$p$$ in $$G$$, iff any automorphism of $$G$$ satisfying property $$p$$, sends $$H$$ to within itself.

In terms of the function restriction formalism
A subgroup property $$p$$ is termed an auto-invariance property if there is a group-closed automorphism property $$q$$, such that we can write the following function restriction expression for $$p$$:

$$p = q \to $$ Function

In other words, a subgroup has subgroup property $$p$$ if every automorphism with property $$q$$, for the whole group, restricts to a function from the subgroup to itself.

This is equivalent to the following function restriction expressions:

$$p = q \to $$ Automorphism

and:

$$p = q \to $$ Endomorphism

In other words, the restriction is automatically guaranteed to be an automorphism of the subgroup.

Equivalence of definitions
The equivalence of definitions follows from the elementary observation: restriction of automorphism to subgroup invariant under it and its inverse is automorphism.

Normal subgroups
The property of normality is an auto-invariance property, where the group-closed automorphism property in question is the property of being an inner automorphism.

Characteristic subgroups
The property of being characteristic is an auto-invariance property, where the group-closed automorphism property in question is the property of being any automorphism.

Weaker metaproperties

 * Stronger than::Endo-invariance property
 * Stronger than::Invariance property
 * Stronger than::Strongly join-closed subgroup property
 * Stronger than::Strongly intersection-closed subgroup property
 * Stronger than::Automorphism-based relation-implication-expressible subgroup property
 * Stronger than::Normalizer-closed subgroup property
 * Stronger than::Centralizer-closed subgroup property
 * Stronger than::Commutator-closed subgroup property