Let $G, H$ be locally compact, $\sigma$-compact metric groups equipped with left Haar measures $m_G, m_H$ respectively. Let $\Phi: G \times H \to G$ be continuous such that $\phi: H \to Aut(G), \, h \mapsto \Phi( \cdot, h)$ is a group homomorphism. Denote $\Phi_h = \phi(h)$. Then, the semidirect product $G \rtimes_\phi H$ is defined as the topological space $G \times H$ equipped with $$(g_1, h_1) \cdot (g_2, h_2) = (g_1 \Phi_{h_1}(g_2), h_1 h_2)$$ for $g_1, g_2 \in G, \, h_1, h_2 \in H$. This is a topological group.
I would like to show that $$dm_{G \rtimes_\phi H}(g,h) = mod(\Phi_h) \,dm_H(h)\, dm_G(g) $$ defines a left Haar measure on $G \rtimes_\phi H$, where $mod$ is of course the modular function $mod: Aut(G) \to \mathbb{R}_{>0}$ defined by $$m_G(\varphi^{-1} B) = mod(\varphi) m_G(B)$$ for Borel sets $B \subseteq G$.
How can I do this without any explicit form for $\Phi_h$? The definition for the semidirect product on wikipedia uses conjugation instead of a general automorphism. Is it obvious from my definition that $\Phi_h$ must be some conjugation?