Weakly procharacteristic subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed weakly procharacteristic if for any automorphism $$\sigma$$ of $$G$$, the following holds: if $$K$$ denotes the closure of $$H$$ under the action of the cyclic group generated by $$\sigma$$, there exists $$g \in K$$ such that $$\sigma(H) = gHg^{-1}$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Intermediately isomorph-conjugate subgroup
 * Weaker than::Intermediately automorph-conjugate subgroup
 * Weaker than::Procharacteristic subgroup

Weaker properties

 * Stronger than::Automorph-conjugate subgroup
 * Stronger than::Weakly pronormal subgroup

Effect of property operators
If $$H$$ is a subgroup of $$G$$ that is weakly procharacteristic in every intermediate subgroup of $$G$$ containing it, then $$H$$ is an intermediately automorph-conjugate subgroup of $$G$$. Conversely, if $$H$$ is intermediately automorph-conjugate in $$G$$, then $$H$$ is weakly procharacteristic in every intermediate subgroup.

Facts

 * Any weakly procharacteristic subgroup of a normal subgroup is weakly pronormal.
 * Weak procharacteristicity is the most general property for which this is true.