It can be proven that for any vector space $V$ the action of $\mathrm{GL}(V)$ on $V \setminus \{0\}$ is transitive, and its stabilizer is $U^* \rtimes GL(U)$, where $U$ is a complement to the subspace spanned by some non-zero vector from $V$. The proof that I found relies heavily on the fact that any non-zero vector can be extended to a basis, so naturally it's a useless tactic for modules.
Can this proposition be generalized to modules or at least some class of modules larger than vector spaces? What proof strategy would you suggest?