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Thanks for the addition. I have edited a little:
Have added hyperlinks (just enclose technical keywords in double square brackets!).
Have changed the notation for the delooping groupoid of $G$ from $G$ to $\mathbb{B}G$.
Adjusted wording a little.
Now it reads like so, but be invited to edit further:
Writing $\mathbb{B} G$ for the category with a single object $\ast$ and the group $G$ as its hom set (i.e. the delooping groupoid of $G$), define a functor $F \colon \mathbb{B}G \to$ Cat to send that single object to the delooping groupoid of $\Gamma$, i.e. $* \mapsto \mathbb{B}\Gamma$ and to send the morphisms $G \to Aut(\Gamma)$ according to the given action of $G$ on $\Gamma$.
Then the delooping of the semidirect product group $\Gamma \rtimes G$ arises as the Grothendieck construction of this functor:
$\mathbb{B}( \Gamma \rtimes G) \simeq \int_{\mathbb{B}G}F$1 to 3 of 3