We work over the complex numbers $ k = \mathbb{C} $.
Let $ F $ be a moduli problem, i.e. a contravariant functor $ F : \text{Sch}/k \rightarrow \text{Sets} $. Suppose that $ M $ is a coarse moduli scheme for the moduli problem $ F $. Also suppose that the objects represented by this moduli problem, i.e. the objects corresponding to $ F(k) \cong M(k) $ have no non-trivial automorphisms.
Is $ M $ also a fine moduli scheme for the moduli problem $ F $?
I am aware of algebraic stacks but I would prefer an answer entirely about schemes, if possible.
I suspect this is false but haven't really been able to come up with a concrete counterexample, even for topological objects like triangles. Thank you for any help.
Edit. For clarity, I will add the definitions I'm using.
A coarse moduli scheme for $ F $ is a scheme $ M $ over $ \mathbb{C} $ and a natural transformation $ t : F \rightarrow \text{Hom} ( - , M) $ such that (1) for every algebraically closed field $ \Omega $ (containing $ \mathbb{C} $) we have $ t : F(\Omega) \rightarrow M(\Omega) $ is a bijection. (2) Given any scheme $ N $ and a natural transformation $ u : F \rightarrow \text{Hom} ( - , N) $, there exists a unique morphism of schemes $ s : M \rightarrow N $ such that $ u = st $.
A fine moduli scheme for $ F $ is a scheme $ M $ and a natural isomorphism $ t : F \rightarrow \text{Hom} ( - , M) $.
Using these definitions, it is clear that if a coarse and fine moduli scheme exist, they are unique upto a unique isomorphism.