I don't think this is the case but I can't find an error in my proof:
Let $N \subset M$ be a submodule of an injective module $M$. Suppose we have maps $f:A \to N$ and $h:A \to B$ and we want to prove that there is some map $g:B \to N$ so that $g \circ h = f$.
Since $M$ is injective consider the induced map $f' : A \to M$. Then there must exist $g':B \to M$ so that $g' \circ h = f'$. But the range of $f'$ is $N$, so that range of $g'$ must be $N$ and thus $g'$ is the desired map.