The Wikipedia page on Representable Functor says:
A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a G-set.
I don't understand how the functor applies to a map on $G$. Suppose we have a group homomorphism $f:G\rightarrow G$ and a functor $F$ from $G$ to $Set$ given by a $G$-set $X$. What is $Ff$?
Thank you very much.
Actually, you have an error in understanding the category $\mathcal{G}$ obtained from $G$.
It is not the full subcategory of $\mathbf{Grp}$ whose only object is $G$: i.e. it is not the category with one object whose arrows are endomorphisms of $G$.
Instead, it is the abstract category with one object whose arrows are elements of $G$. The product of arrows is given by the product of $G$.
This construction works with monoids too. Conversely, every category with one object is equivalent to one constructed from a monoid in this way.
In any category $\mathcal{C}$, one can see that the set $\hom_\mathcal{C}(X, X)$ of endomorphisms of an object $X$ can be given a monoid structure, using the category's product law. In fact, for the object of $\mathcal{G}$, this monoid is equal to $G$.
Any functor $\mathcal{C} \to \mathcal{D}$ can be seen to induce monoid homomorphisms $\hom_{\mathcal{C}}(X,X) \to \hom_{\mathcal{D}}(FX, FX)$. Your proposition of interest is a special case of this fact, using the appropriate characterization of group actions.