When can we extend abelian group functors from a subcategory of $\textbf{Ring}$ to the whole category?

Question

Let $C\subset\textbf{Ring}$ be a subcategory (not necessarily full).

Let $G$ be a covariant abelian group functor $G : C\rightarrow\textbf{Ab}$.

What conditions do we need to impose so that $G$ admits an extension to an abelian group functor on $\textbf{Ring}$? Under what conditions is this extension unique?

For example, it seems that if:

1. $C$ has an initial object, which is also initial in $\textbf{Ring}$ (ie, $\mathbb{Z}$ is an initial object of $C$),
2. $G$ is representable by a ring $R\in C$, and
3. the coproduct $R\otimes R$ in $C$ is also the coproduct in $\textbf{Ring}$,

then $G$ should extend uniquely to $\textbf{Ring}$.

Am I missing anything? Are there more general conditions? (Is there a reference for this?)