Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ and $\text{Ad}^H:H \to \text{Aut}_\text{Lie}(\mathfrak{h})$. Moreover, let $\phi:G \to \text{Aut}_\text{Grp}(H)$ be a Lie group action. My question is now, is there a canonical way to construct the adjoint action $\text{Ad}^{H \rtimes_\phi G}:H \rtimes_\phi G \to \text{Aut}_\text{Lie}(\mathfrak{h} \rtimes_{\text{Lie}(\phi)} \mathfrak{g})$ on the semi-direct product from the actions $\text{Ad}^G$ and $\text{Ad}^H$?
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