Let $R$ be a commutative ring with unity. Let $$ 0\to A\stackrel{f}{\to} B\stackrel{g}{\to} C\to 0 $$ be an exact sequence of $R$-modules. This induces an exact sequence for each $R$-module $M$: $$ 0\to\mathrm{Hom}_R(C,M)\stackrel{\bar{g}}{\to}\mathrm{Hom}_R(B,M)\stackrel{\bar f}{\to}\mathrm{Hom}_R(A,M), $$ here $\bar f$ sends $h$ to $h\circ f$ and $\bar g$ is similarly defined.
Question:
Find an example of $0\to A\to B\to C\to 0$ and $M$ such that $\mathrm{Hom}_R(B,M)\to\mathrm{Hom}_R(A,M)$ is not surjective.
Up to isomorphisms, we can just consider $0\to A\stackrel{f}{\to} B\stackrel{g}{\to} B/A\to 0$, where $A$ is a submodule of $B$, and $f$, $g$ are natural homomorphisms. This simplifies the problem a little bit but I am still not sure how to find such an example.