Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ is closed under biproducts (see [this question][1]).

If we take a cofibrant object $X$ and a fibrant object $Y$ then there is a natural isomorphism
$$
 \operatorname{Ho}\mathcal C(X,Y) \cong \mathcal C(X,Y)/\sim
$$
where $\sim$ is the homotopy relation. Is this always a group isomorphism?

  [1]: http://mathoverflow.net/questions/44047/localizing-an-arbitrary-additive-category