Automorphism-based relation-implication-expressible subgroup property

Statement
Suppose $$a$$ and $$b$$ are two group-closed properties of automorphisms from a group to itself. Let $$R_a$$ denote the following relation between subgroups: two subgroups $$H, K \le G$$ satisfy $$R_a$$ if there is an automorphism of $$G$$ satisfying property $$a$$ and sending $$H$$ to $$K$$. Analogously, define $$R_b$$. The automorphism-based relation-implication-expressible subgroup property for $$a$$ and $$b$$ is the property with relation implication expression:

$$p = R_a \implies R_b$$.

In other words, $$H$$ satisfies property $$p$$ in $$G$$ if whenever $$(H,K)$$ satisfy $$R_a$$ (i.e., there is an automorphism with property $$a$$ sending $$H$$ to $$K$$), $$(H,K)$$ also satisfy property $$R_b$$ (i.e., there is an automorphism with property $$b$$ sending $$H$$ to $$K$$).

Examples
Here are some examples:


 * Characteristic subgroup: Here, the two properties are automorphism and identity map.
 * Normal subgroup: Here, the two properties are inner automorphism and identity map.
 * Automorph-conjugate subgroup: Here, the two properties are automorphism and inner automorphism.

Stronger metaproperties

 * Weaker than::Auto-invariance property

Weaker metaproperties

 * Stronger than::Centralizer-closed subgroup property
 * Stronger than::Normalizer-closed subgroup property