Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod $ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space.

Can you find sufficient conditions so that $F$ is the restriction functor relative to $f:B \to A$?

I suspect that a "nice" condition to start from is monoidality. Indeed, in the case in which the rings are the group rings of finite groups, by Tannaka reconstruction you could reconstruct the morphism.

My motivation: I had to find an $\mathbb{R}$-algebra injective morphism from $\mathbb{C} \to \mathbb{R}[u]/(u^2+1) ^m$. It's really easy with Jordan form theory to give a complex structure on a module for the right hand ring, but constructing the actual morphism it's not easy at all, and some non trivial ad-hoc argument is needed to make " reconstruction ".