Automorph-approximate left residual

Statement
Suppose $$p,q$$ are two (not necessarily distinct) subgroup properties. The automorph-approximate left residual of $$p$$ by $$q$$ is the subgroup property $$r$$ defined as follows:

A subgroup $$H$$ of a group $$G$$ satisfies property $$r$$ in $$G$$ if, for every group $$K$$ containing $$G$$ such that $$G$$ satisfies property $$q$$ in $$K$$, there is automorphism $$\sigma$$ of $$G$$ such that $$\sigma(H)$$ satisfies $$p$$ in $$K$$.