= Solution
Restriction of scalars makes every $S^{-1}A$-module an $A$-module, and multiplication by $s\in S$ is invertible with inverse multiplication by $1/s$.
Conversely, suppose each multiplication map $s_M:m\mapsto sm$ is an automorphism of the $A$-module $M$. Define
$$
\frac as\cdot m=a\,s_M^{-1}(m).
$$
The universal property of localization shows that this is well-defined and gives the unique $S^{-1}A$-module structure extending the $A$-action. These constructions are inverse.
Under this structure the natural map $M\to S^{-1}M$ has inverse
$$
\boxed{\frac ms\longmapsto s_M^{-1}(m),}
$$
so it is an isomorphism.
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