Let $S\subset R$ be rings. Denote $R^n$ the free $R-$module of rank $n$, and $S^n$ a $R-$submodule. Let $\alpha, \beta: R^n\rightarrow R^n$ be $R-$module automorphisms such that $\alpha(S^n), \beta(S^n)\subset S^n$.
Question: is it true that $\alpha(S^n)\cong \beta(S^n)$ as $S-$modules?
As $S \subseteq R$ any $R$-module homomorphism is also an $S$-module homomorphism. So if $\alpha$ is an $R$-module isomorphism then it's also an $S$-module isomorphism. This means $S^n$ and $\alpha(S^n)$ are isomorphic as $S$-modules. The same is true of $\beta$, so $\alpha(S^n) \simeq S^n \simeq \beta(S^n)$.