Let $\mathcal{G}=(A\rightrightarrows X)$ be a groupoid.
Here $X={\rm Ob}(\mathcal{G})$, $A={\rm Ar}(\mathcal{G})$,
and we have 5 maps:
$s,t\colon A\to X$ (the source and the target, surjective),
$m\colon A\times\_X A\to A$ (multiplication of composable arrows),
${\rm id}\colon X\to A$ ($x\mapsto{\rm id}\_x$, injective),
and $i\colon A\to A$ ($a\mapsto a^{-1}$),
satisfying the usual axioms.
I say that my groupoid is connected if for any two objects $x,y\in X$ there exists an arrow $a\colon x\to y$.


Assume that a finite group $\Gamma$ acts on $\mathcal{G}$, i.e., it acts on $X$ and $A$ such all the 5 maps are $\Gamma$-equivariant.
We say that $\mathcal{G}$ is a $\Gamma$-groupoid.


Now I want to construct a fibered category (gerbe) $\mathbb{G}$ over the category (site) of finite $\Gamma$-sets,
starting from a connected $\Gamma$-groupoid $\mathcal{G}$.
In other words, for any finite $\Gamma$-set $S$, I want to construct a groupoid $\mathbb{G}(S)$,
and for a morphism $S\to T$ of finite $\Gamma$-sets, I want to define a restriction functor $\mathbb{G}(T)\to \mathbb{G}(S)$.
How can I do that?
I could not find this in Giraud's book.



Constructing a stack (gerbe) from a connected groupoid
The thing that Simon gives in his first comment is just a prestack, it needs to be stackified. You can do this by letting $\mathbb{G}(S)$ be the groupoid of principal $\mathcal{G}$-bundles in $\Gamma Set$, in other words, $\Gamma$-equivariant $\mathcal{G}$-bundles in Set.


In more detail, I'm assuming you are considering the topology on $\Gamma Set$ to be the one where a covering family is a surjective map. Then we define a principal $\mathcal{G}$-bundle over a $\Gamma$-set $S$ to be a surjective map of $\Gamma$-sets (all maps from now on will be $\Gamma$-equivariant), $\pi\colon P\to S$, a map $b\colon P\to X$ and an *action* map $a\colon P\times\_{X,s} A \to P$ such that $\pi(a(p,f)) = \pi(p)$ (the action is fibrewise); in addition, we demand that the map $P\times\_X A \to P\times\_S P$ is an isomorphism.


Additionally, we demand there is a cover $j\colon T\to S$ and a map $\sigma\colon T \to P$ such that the pulled back bundle $T\times\_S P \to T$ is isomorphic to the pullback of the canonical $\mathcal{G}$-bundle $ t\colon A \to X$ (the action is by composition in the groupoid) along $b\circ \sigma\colon T \to P \to X$ 


The groupoid $\mathbb{G}(S)$ is that of principal $\Gamma$-bundles with bundle maps between them (they preserve fibres and actions, including the data of the map to $X$). The functor $\mathbb{G}(S) \to \mathbb{G}(T)$ is given by pullback.


