On page 69 in Mumford's Abelian Varieties book, he says:
"Let $G$ be a finite group acting on a variety $X/k$, and let $X\stackrel{\pi}{\longrightarrow} Y := X/G$ be the quotient. Let $\mathcal{F}$ be a coherent sheaf on $Y$. For any $g\in G$, we have $\pi\circ g = \pi$, and so we deduce that there is a natural automorphism $g^* : \pi^*(\mathcal{F})\rightarrow\pi^*(\mathcal{F})$ over the action of $g$ on $X$. Thus, $G$ acts on $\pi^*(\mathcal{F})$ in a manner compatible with its action on $X$. By a coherent $G$-sheaf on $X$, we shall mean a coherent $\mathcal{O}_X$-module on which $G$ acts in a way compatible with its action on $X$."
In a paper by Ellingsrud/Lonsted An Equivariant Lefschetz Formula for Finite Reductive Groups, page 254, they write:
"When $G$ acts on $X$ over $k$ we call $X$ a $(G,k)$-scheme, and a $G$-linearized $\mathcal{O}_X$-module (c.f. [Mumford] p. 110) will be called a $(G,\mathcal{O}_X)$-module. For $X = \text{Spec }A$ we just say $A$-module and $(G,A)$-module. The latter is equivalent to $A[G]$-right module of finite type, where $A[G]$ is the group ring of $G$ with coefficients in $A$."
(They refer to Mumford p.110, but p.110 seems to just be a translation of p.69 into group scheme language)
These definitions are confusing to me. For example, if we take $X = \text{Spec }A$ equipped with an action of $G$, and $\mathcal{F} = M$, an $A$-module, then how do we want $G$ to act on $M$ in order to call it a $G$-sheaf?
By working through the definition in Mumford, it a-priori seems that we would want $G$ to act on $M$ as automorphisms of abelian groups such that for all $g\in G, m\in M, a\in A$ we have $$g(a\cdot m) = g(a)\cdot g(m)$$ What's confusing me, however, are two things:
The last quoted sentence of Mumford seems to imply that $G$ is acting on the $\mathcal{O}_X$-module $\mathcal{F}$, which sort of suggests that it is acting as $\mathcal{O}_X$-module automorphisms, which would correspond to the rule $g(a\cdot m) = a\cdot g(m)$. This would contradict what I understood from the definition.
In the quoted text from Ellingsrud/Lonsted, they say that viewing $M$ as a $(A,G)$-module is the same as giving it the structure of an $A[G]$-module. Now, $A[G]$-modules are definitely just $A$-modules $M$ equipped with an $A$-linear action of $G$, which again suggests the rule $g(a\cdot m) = a\cdot g(m)$.
I've also read some other articles on $G$-sheaves and "equivariant sheaves", but many of them use somewhat difficult to unravel language.
In any case, I'm leaning towards the definition where $g(a\cdot m) = g(a)\cdot g(m)$, and I'm willing to accept that Mumford's last sentence is just poorly phrased, but I remain confused about why Ellingsrud/Lonsted would say that $G$-sheaves are the same as $A[G]$-modules. Am I misunderstanding some common terminology here?
