Conjugacy-closed sub-APS

Definition
A sub-APS $$H$$ of an APS of groups $$(G,\Phi)$$ is termed conjugacy-closed if for every $$n$$, $$H_n$$ is a conjugacy-closed subgroup of $$G_n$$: in other words, whenever two elements of $$H_n$$ are conjugate in $$G_n$$, they are also conjugate in $$H_n$$.