David Roberts already gave a nice answer, which you accepted. However, let me give you another perspective. Your groupoid $\mathcal{G}$ has an action of $\Gamma,$ i.e. is a groupoid object in $Set^{B\Gamma}\cong \Gamma-Set$ (where $B\Gamma$ is $\Gamma$ viewed as a one object category). So $$\mathcal{G} \in Gpd\left(Set^{B\Gamma}\right) \cong Gpd^{B\Gamma}.$$ I.e., your groupoid $\mathcal{G}$ is the same data as a funtor $\mathbb{G}:B\Gamma \to Gpd$. Explicitly, it sends the one object to the underlying groupoid in sets of $\Gamma,$ and $g \in \Gamma$ gets sent to the functor $\mathcal{G} \to \mathcal{G}$ induced by the action.

Now, you wanted to arrive at a gerbe over the large site $\Gamma-Set$ with surjections as covers. Well, this is the canonical topology on the topos $\Gamma-Set$, and sheaves over $\Gamma-Set$ with respect to this topology are equivalent to $\Gamma-Set$ itself, so, stacks over this site, are equivalent to the bicategory of (weak) functors $B\Gamma \to Gpd$, of which $\mathbb{G}$ *is* an example. So it corresponds canonically to a stack on $\Gamma-Set$. Explicitly, given a $\Gamma$-set $X,$ $X$ may viewed as a functor $X:B\Gamma \to Set$ and hence also as a functor $X^{id}:\Gamma \to Gpd,$ where we view a set as a groupoid with all identity arrows. Then the stack on $\Gamma-Set$ that $\mathbb{G}$ corresponds to, sends $X$ to the groupoid of maps $Hom\left(X^{id},\mathbb{G}\right).$ Provided that $\pi_0\mathcal{G}=*$ this stack is a gerbe.

If you prefer to get this stack as a fibered category, there is another approach. Consider the Grothendieck construction of $\mathbb{G}$ $$\pi:\int_{B\Gamma} \mathbb{G} \to B\Gamma.$$ It can be canonically identified with the action groupoid $\Gamma \ltimes \mathcal{G} \to B\Gamma.$ (See e.g. the "generalized action groupoid" construction in http://arxiv.org/abs/1011.6070, or see the homework assignment I gave my topos theory class: http://people.mpim-bonn.mpg.de/carchedi/HW1.pdf). Anyway, this is a fibered category over $\Gamma$ describing the gerbe associated to $\mathcal{G}.$ The fibered category over $\Gamma-Set$ it corresponds to is "sheaves over $\int_{B\Gamma} \mathbb{G}$ with the induced Grothendieck topology" which is easily seen to be the same as $Set^{\Gamma \ltimes \mathcal{G}},$ (since the topology becomes the canonical topology again) which becomes a fibered category over $Set^{\Gamma}$ via the functor $$\pi_!:Set^{\Gamma \ltimes \mathcal{G}} \to Set^{\Gamma},$$ where $\pi_!$ is left adjoint to the restriction functor.

2more comments