Let $P$ be a poset on which a partially ordered group $G$ acts by monotone bijections. I call these guys $G$-posets. If I'm not wrong, the semidirect product of two partially ordered groups remains partially ordered. It is then possible to ask:
If $P$ is a $G$-poset and $Q$ a $H$-poset, is there a way to find a third poset that I call for now $P\rtimes_\phi Q$ which has an action of $G\rtimes_\phi H$ (for a given $\phi : H\to Aut(G)$)?
Is there a way to promote this construction to a functor $$ G\text{-Pos}\times H\text{-Pos} \to (G\rtimes_\phi H)\text{-Pos} $$
Given a $(G\rtimes_\phi H)$-poset $P$ is there a way to find when there are $P_G, P_H$ such that $P\cong P_G\rtimes_\phi P_H$?