Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between them. There is a free functor $G \colon \mathbf{Set} \to \mathbf{Vect}$, left adjoint to the forgetful functor, given by $G(X) = \mathbb{C}^X$ and $G(f)(\phi)(y) = \sum_{f(x)=y} \phi(x)$.

Is any functor $\mathbf{Set} \to \mathbf{Vect}$ of the form $H \circ G \circ F$ for $F \colon \mathbf{Set} \to \mathbf{Set}$ and $H \colon \mathbf{Vect} \to \mathbf{Vect}$?

A priori one might expect lots more functors, but I'm having a hard time coming up with any. On the other hand, functoriality seems to keep $F$ from "creating chaos" to "mess up freeness" (sorry that I can't explicate this feeling better). To keep it simple, let's keep things finite(-dimensional), and not consider anything about other base fields, or monoidal structure.