Let $\pi:P\to B$ is a $G$-principal bundle, which means $G$ acts on $P$ freely and $\pi$ is a locally trivial fibration. Here is a well-known theorem:

**THeorem:** The inverse image functor $\pi^{*}$ gives an equivalence from $\mathbf{Sh}_G(P)$ to $\mathbf{Sh}(B)$, and the inverse functor is given by $\pi_* ^G$.

Let me explain some notations: an object in $\mathbf{Sh}_G(P)$ is a pair $(\mathcal{F},\alpha)$, where $\alpha:p^{*}\mathcal{F}\simeq a^{*}\mathcal{F}$, and satisfies the cocycle condition. Here $p:G\times P\to P$ is the projection, and $a:G\times P\to P$ is the action.
The functor $\pi_{*}^G$ is given by assigning open subset $U$ to $\mathcal{F}(\pi^{-1}(U))^G$.

We need to show that:

- $\pi^{*}$ is fully-faithful, i.e., for any two sheaves $\mathcal{F}$ and $\mathcal{G}$, $Hom(\mathcal{F},\mathcal{G})\simeq Hom_G(\pi^{*}\mathcal{F},\pi^{*}\mathcal{G})$.
- $\pi^{*}$ is essentially surjective, i.e., for any $G$-equivariant sheaf $\mathcal{H}$, there exists an isomorphism $\mathcal{H}\simeq\pi^{*}\pi_{*}^G\mathcal{H}$.

(1) follows from $\pi_{*}^G\circ\pi^{*}\simeq Id$, which is relatively easy.

For (2), I have checked when $B$ is a point. For general case, I can reduce it to the following isomorphism: $\pi_{*}(\mathcal{H})_b^G\simeq\Gamma\left(\pi^{-1}(b), \mathcal{H}|_{\pi^{-1}(b)}\right)^G$, where $b\in B$.

I know for nice group $G$, for example when $G$ is compact, this is really an isomorphism, because we have base change. For general group, I have no idea at all.

Who can help to finish this proof? Or give some other idea?