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 $K$.