Sub-APS of groups

Definition
Let $$(G,\Phi)$$ be an APS of groups. A sub-APS $$H$$ of $$G$$ is, for every $$n$$, a subgroup $$H_n$$ of $$G_n$$ such that $$\Phi_{m,n}(g,h) \in H_{m+n}$$ whenever $$g \in H_m, h \in H_n$$. Thus, $$H$$ can be viewed as an APS of groups, in its own right.