Let $R$ be a ring with unit and $E$ and $F$ be two finitely generated projective left modules of $R$. Let $I$ and $J$ be submodules of $E$ and $F$ respectively.
If we have an $R$-homomorphism $\phi: I\to J$, could we always extend $\phi$ to an $R$-homomorphism $\widetilde{\phi}: E\to F$? If not, is there an simple counterexample?
I believe counterexamples exist even in abelian groups. So, let $R=\mathbb{Z}$ so that everything is simply an abelian group.
Let $E=F=\mathbb{Z}$, $I = 4\mathbb{Z}$, and $J= 2\mathbb{Z}$. Certainly $E=F$ is f.g. projective (free, even). Then $\phi: I \to J$ by $\phi(x) = \frac{x}{2}$ is a group homomorphism. However, it cannot extend to $\tilde{\phi}: \mathbb{Z} \to \mathbb{Z}$ since such a $\tilde{\phi}$ would have to map $4$ to $2$, and no group homomorphism on $\mathbb{Z}$ does that.