A question about the definition of the module structure on sections of a local system

A question about the definition of the module structure on sections of a
local system

For me, a local system on a topological space $X$ is a locally constant
sheaf on $X$, i.e., a sheaf of sets $\mathscr{F}$ for which there is an
open cover $X=\bigcup_i U_i$ so that
$\mathscr{F}_i\cong\underline{\mathscr{F}(U_i)}_{U_i}$, the constant sheaf
on $U_i$ associated to the set $\mathscr{F}(U_i)$.
The local systems I'm interested in arise as follows (and perhaps they all
arise this way, maybe under some additional connectedness assumptions on
$X$). Let $G$ be a group acting (on the left, let's say) freely and
properly discontinuously on $X$ (I'm not sure if the definition of the
latter term is universally agreed upon, but the condition I want is that
each $x\in X$ admits an open $x\in U\subseteq X$ such that $gU\cap
U=\emptyset$ for $g\neq 1$) and $V$ an $R$-module (for some commutative
ring $R$) with an $R$-linear action of $G$. Then the natural projection
$q:G\setminus(X\times V)\rightarrow G\setminus X$ (here I regard $V$ as
having the discrete topology and let $G$ act on $X\times V$ diagonally) is
then a covering space (because of the assumption on the action of $G$ on
$X$), and all the fibers are in bijection with $V$. So the sheaf of
continuous sections $\mathscr{F}_V$ on $G\setminus X$ is going to be
locally on $X$ isomorphic to the constant sheaf $\underline{V}_X$. Now, my
understanding is that something like this is called a local system of
$R$-modules, and my question is:
What is the $R$-module structure on the set of sections $\mathscr{F}_V(U)$
for $U\subseteq X$ open?
The only thing that I can think of (and I guess I'm motivated by the
vector space structure on the sections of a vector bundle) is that for
each $x\in X$, the restriction of $X\times V\rightarrow G\setminus(X\times
V)$ to $\{x\}\times V$ is a bijection onto the fiber over $Gx$ of $q$. So
one can make the fiber into an $R$-module by declaring that this bijection
should be an isomorphism of $R$-modules. Then the $R$-module structure on
$\mathscr{F}_V(U)$ would be defined pointwise (like for vector bundles),
i.e., if $s,t\in\mathscr{F}_V(U)$, so that $s(u),t(u)\in q^{-1}(u)$, then
$(s+t)(u)=s(u)+t(u)$, the addition taking place in the fiber $q^{-1}(u)$,
and likewise, for $r\in R$, $(rs)(u)=rs(u)$, the scalar multiplication
taking place in $q^{-1}(u)$.
I think that with this definition, if $U$ is an open set of $X$ over which
the covering $G\setminus(X\times V)\rightarrow G\setminus X$ can be
trivialized, then the induced isomorphism
$\mathscr{F}_V\vert_U\cong\underline{V}_U$ will be one of sheaves of
$R$-modules, which suggests I have the right definition...but I haven't
checked the details. Do I have the right definition of the $R$-module
structure?