Automorphism-based relation-implication-expressible subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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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 $H,K\leq G$ 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.

Relation with other metaproperties

Stronger metaproperties

Auto-invariance property

Weaker metaproperties