I think that the following question is conceivably research-level, but it arose just as a curiosity, not part of any serious research. Is it reasonable to ask it on MO? (I include the specific question in case it affects your answer, but I am not intending to ask the question in this inappropriate forum.)
Suppose that $V_1$ and $V_2$ are vector spaces (over $\mathbb C$, say) with the same underlying set $X$. Suppose that the set of permutations of $X$ that are linear automorphisms of $V_1$ is the same as the analogous set for $V_2$. Then are $V_1$ and $V_2$ isomorphic?
EDIT: Stefan Kohl points out that it doesn't matter how the question arose, so probably I should clarify that I was asking because I don't know if a random curiosity is research-level—as opposed to something that arose in my research, which more or less by definition is (hopefully!).