Lie algebra of a semi-direct product of Lie groups

Question

Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ and suppose $G$ acts on $H$ by automorphisms, i.e. there exists a Lie group homomorphism $\phi:G\to Aut(H)$. Consider the semidirect product $G\ltimes H$, which is again a Lie Group. Is its Lie algebra the semidirect product Lie algebra of $\mathfrak{g}$ and $\mathfrak{h}$?