$\require{AMScd}$When regarded as a functor, the semidirect product projection $p : G\ltimes_\phi H \to G$ with respect to a representation $\phi : G\to Aut(H)$ is a Grothendieck fibration: it corresponds to the functor $BG \to Cat$ "choosing $\phi$" (the unique object of $BG$ is sent to $H$, clearly a category, and $\phi$ is the action on morphisms).
But then, it's a Conduché functor, which is very nice, because there is an adjunction $$p^* : Cat/BG \leftrightarrows Cat/B(G\ltimes_\phi H) : \Pi_p$$ where $p^*$ is pulling back along $p$, and $\Pi_p$ is dependent product.
Computing $\Pi_p$ is notoriously difficult: I have no particular idea how to proceed, and if there's some hope to write down an explicit description. Can it be done?
The only "simplification" that came to my mind is that given a group $A$, the slice $Cat/A$ is the comma category $q/A$, where $q : Cat \to Grp$ is the composition of functors $$ \begin{CD} Cat @>k>> Gpd @>\dot q>> Grp \end{CD} $$ where $k$ is groupoidification, left adjoint to $Gpd\hookrightarrow Cat$, and $\dot q$ is the functor taking a groupoid $\mathcal G$ to the pushout of $$ \begin{CD} \mathcal G_0 @>>> \mathcal G \\ @VVV@.\\ 1 @. \end{CD} $$ but it doesn't seem that the computation is simplified by this remark (on the contrary...).