I have a small gap when proving the titled proposition, namely, given a finite constant group scheme $G$ over a field $k$, and let $U:=Spec(A)$ be an affine $k$ scheme which $G$ act on, then quasi-coherent sheaf on $\mathfrak{X}:=[Spec(A)/G]$ corresponds to an A-module M equipped with a group homomorphism $G\to End_A(M)$.
The key ingredient in the proof is the identification of $U\times_{[U/G]}U\cong G\times U$. To prove this, on objects we identify $(u_1,u_2,g)$ with $(g,u_1)$; and on morphisms, since $U\times_{[U/G]}U$ is representable by algebraic spaces, its stabilizers are trivial, and thus the coincidence.
Then since given a etale presentation $U\to \mathfrak{X}$, the category of quasi-coherent sheaves on $\mathfrak{X}$ is equivalent to the category of quasi-coherent sheaves on $U$ with descent datum. Apply this to $U,\mathfrak{X}$ mentioned in the first paragraph, a quasi-coherent sheaf $\mathfrak{M}$ on $\mathfrak{X}$ is an $A$-module $M$, together with an isomorphism $\alpha: p_1^* M \cong p_2^* M$, and the cocycle condition: $p_{23}^* \alpha\circ p_{12}^* \alpha=p_{13}^* \alpha$. Since $G$ is a finite constant group scheme, $G=Spec (\prod k)$. So by the identification $U\times_{[U/G]}U\cong G\times U$, we may set $p_1:G\times U\to U, (g, u)\to u, p_2:G\times U\to U, (g,u)\to gu$. On the level of rings, $p_1:A\to A\otimes_k \prod k=\prod A, a\to (a,a,...,a)$ and $p_2:A\to \prod A, a\to (g_1 a, g_2 a,..., g_n a)$. Then we can rewrite the map $\alpha: \oplus M\to \oplus M\otimes_{A,g:A\to A} A$. But I am unclear how this can give a map $G\to End_A(M)$. I have the same issue with the following as well: I know we need to cocycle condition to show that the map is a homomorphism, and I get stuck at a similar place as above.
