Right notion of $G$-equivariant $A$-modules which is equivalent to $G$-equivariant quasicoherent sheaves over $X=\operatorname{Spec} A$.

As I mentioned in my previous post, I have been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on logarithmic schemes. While editing that post, a second question came up, which I thought was sufficiently independent to deserve its own post. So here is the post.$\def\Spec{\operatorname{Spec}}\def\QCoh{\operatorname{QCoh}}\def\pr{\operatorname{pr}}\def\Grp{\operatorname{Grp}}\def\Aff{\operatorname{Aff}}\def\CRing{\operatorname{CRing}}$

Context

The paper contains the following proposition

Proposition 4.6 Let $\mathscr X$ be a fibred category over $(\Aff)$ with an fpqc atlas $U\to \mathscr X$. Then the pullback $R=U\times_{\mathscr X}U$ is represented by a scheme, and we obtain an fpqc groupoid $R\rightrightarrows U$. Write $\QCoh(R\rightrightarrows U)$ for the category of quasicoherent sheaves $F$ on $U$ equipped with descent data $\pr_1^*F\simeq\pr_2^*F$. Then there is an additive equivalence between $\QCoh\mathscr X$ and $\QCoh(R\rightrightarrows U)$.

which can be used to prove the following:

Proposition 4.9 (paraphrased) If $X=\Spec A$ is an affine scheme and $G$ is an affine group scheme over $\mathbb Z$ acting on $X$, then there is an equivalence of tensor categories between the quasi-coherent sheaves over the fpqc quotient $[X / G]$ and the $G$-equivariant quasi-coherent sheaves on $X$.

The problem (?)

Now, their actual proposition 4.9 says "$G$-equivariant $A$-modules" instead of "$G$-equivariant quasi-coherent sheaves on $X$":

Proposition 4.9 (actual version) If $X=\Spec A$ is an affine scheme and $G$ is an affine group scheme over $\mathbb Z$ acting on $X$, then there is an equivalence of tensor categories between the quasi-coherent sheaves over the fpqc quotient $[X / G]$ and the $G$-equivariant $A$-modules.

where their notion of "$G$-equivariant $A$-module" is the following:

If $G$ is an affine group scheme over $\mathbb Z$ acting on a ring $A$, we define an equivariant $A$-module as an $A$-module with an action of $G$, such that for every ring $R$, every $a\in (A\otimes_{\mathbb Z}R), m\in (M\otimes_{\mathbb Z}R)$ and $g\in G(R)$, we have $(ga)(gm)=g(am)$.

I'm wondering if this definition really makes their version of proposition 4.9 true.

Thoughts on how to resolve the issue

Since $X=\Spec A$ has an action of $G$, then $A$ has a coaction of $\mathcal O(G)$, and, as far as I understand, the category of $G$-equivariant quasi-coherent sheaves on $X$ is equivalent to the category of $\mathcal O(G)$-comodules $M$ with a $\mathcal O(G)$-coequivariant multiplication $A\otimes_{\mathbb Z} M\to M$. Let's call these "$\mathcal O(G)$-coequivariant $A$-modules".

So, my concrete questions are the following:

• is their notion of $G$-equivariant $R$-modules also equivalent to the notion of $G$-equivariant quasi-coherent sheaves over $X$? this would mean that their version of proposition 4.9 really is true, and that their definition is equivalent to that of $\mathcal O(G)$-coequivariant $A$-modules. Or...
• did they simply use the wrong definition, and they should have used $\mathcal O(G)$-coequivariant $A$-modules instead?

I'm inclined towards the second option, simply because I haven't been able to take a quasi-coherent $G$-equivariant sheaf on $X$ and produce a "$G$-equivariant $A$-module" in their sense. In fact, I don't even know how to produce an action of $G$ on $A$, which to my understanding should be a natural transformation $G\times\tilde A\to\tilde A$, (where $\tilde A\colon\CRing\to\CRing$ is the functor $R\mapsto(A\otimes_{\mathbb Z}R)$) such that for every commutative ring $R$, the map $G(R)\times(A\otimes_{\mathbb Z}R)\to(A\otimes_{\mathbb Z}R)$ is an action of $G(R)$ by ring automorphisms. The action of $G$ on $X=\Spec A$ naturally produces a coaction $A\to\mathcal O(G)\otimes_{\mathbb Z}A$, and this doesn't look a bit like an action $G\times\tilde A\to\tilde A$.

Still, I might be wrong, and there may be a way to translate between these two definitions, which I haven't been able to find.

Thanks in advance.