Automorph-approximate left transiter

Definition
The automorph-approximate left transiter of a subgroup property $$p$$ is defined as the following subgroup property $$r$$: $$H$$ satisfies property $$r$$ in $$G$$ if, whenever $$G$$ satisfies property $$p$$ in some larger group $$K$$, there exists an automorphism $$\sigma$$ of $$G$$ such that $$\sigma(H)$$ satisfies property $$p$$ in $$K$$.