SETUP: Let $G$ be a group such that $G= N \times_\varphi H$ semi-direct product of $H$ and $N$ with respect to the action $\varphi$. Denote: $$p(G):= \max\{ p \in \mathbb{N}^*\: | \: \mathbb{R}^p \cong K < G \}$$
I know that if $ \varphi = id $ then $ p(G) = p(N)+p(H) $ and that there are some conditions on $\varphi$ such that $p(N \times_\varphi H) = p(N \times H)$.
My question is there, in general, a formula to calculate directly $p(N \times_\varphi H)$ in the function of $p(N)$, $p(H)$, and a constant related to $\varphi$?