In [Lam, Lectures on Modules and Rings] there is the following proposition (15.20):
Let $M$ be finitely generated projective over the Quasi Frobenius ring $R$, and $A,B$ be submodules of $M$. Then any $R$-isomorphism $h : A \to B$ extends to an $R$-automorphism of $M$.
However, in the proof I don't see why the projectivity of $M$ and the property of being Quasi Frobenius of $R$ is essential here (I see that they are used, but I think that they can be replaced by more natural assumptions). If I am not mistaken, the following more general statement should also be true (by the same proof):
Let $M$ be an injective module of finite length over a ring $R$, and $A,B$ be submodules of $M$. Then any $R$-isomorphism $h : A \to B$ extends to an $R$-automorphism of $M$.
Can anyone confirm that? Thank you in advance!