Denote respectively by $\text D(\mathcal C)$ and $\text D(\mathbb Z)$ the derived categories attached to the category $\mathcal C$ of finite abelian groups and to the category of all abelian groups.
Does a right derived bifunctor $$ \text{RHom}_{\mathcal C}:\text D(\mathcal C)^{\text{op}}\times\text D(\mathcal C)\to\text D(\mathbb Z) $$ of the bifunctor $\text{Hom}_{\mathcal C}$ exist?
The setting is that of the book Categories and Sheaves by Kashiwara and Schapira.
Here is a reminder of the definition of a right derived bifunctor
$$
\text{RHom}_{\mathcal C}:\text D(\mathcal C)^{\text{op}}\times\text D(\mathcal C)\to\text D(\mathbb Z)
$$
of $\text{Hom}_{\mathcal C}$: Let
$$
\text{Hom}^\bullet_{\mathcal C}:\text K(\mathcal C)^{\text{op}}\times\text K(\mathcal C)\to\text K(\mathbb Z)
$$
be the triangulated bifunctor induced by $\text{Hom}_{\mathcal C}$ at the homotopy category level, and let
$$
Q:\text K(\mathcal C)^{\text{op}}\times\text K(\mathcal C)\to\text D(\mathcal C)^{\text{op}}\times\text D(\mathcal C),
$$
and $Q':\text K(\mathbb Z)\to\text D(\mathbb Z)$ be the localization functors. A left Kan extension $\text{RHom}_{\mathcal C}$ of $Q'\circ\text{Hom}^\bullet_{\mathcal C}$ along $Q$ (assuming that such exists) is a right derived bifunctor for $\text{Hom}_{\mathcal C}$ if, for any functor $G:\text D(\mathbb Z)\to\mathcal A$, the bifunctor $G\circ\text{RHom}_{\mathcal C}$ is, in a natural way, a left Kan extension of $G\circ Q'\circ\text{Hom}^\bullet_{\mathcal C}$ along $Q$.
Here is an equivalent definition: Assume that for all $X,Y$ in $\text K(\mathcal C)$ the colimit $$ \operatorname*{colim}_{U\to X,Y\to V}\ (Q'\circ\text{Hom}^\bullet_{\mathcal C})(U,V), $$ where $U\to X$ and $Y\to V$ run over all the quasi-isomorphisms in $\text K(\mathcal C)$ (with $X$ and $Y$ fixed), exists in $\text D(\mathbb Z)$, and denote this colimit by $\text{RHom}_{\mathcal C}(X,Y)$. Then $\text{RHom}_{\mathcal C}$ is a right derived bifunctor for $\text{Hom}_{\mathcal C}$ if, for any functor $G:\text D(\mathbb Z)\to\mathcal A$, the colimit $$ \operatorname*{colim}_{U\to X,Y\to V}\ (G\circ Q'\circ\text{Hom}^\bullet_{\mathcal C})(U,V) $$ exists in $\mathcal A$ and coincides with $G(\text{RHom}_{\mathcal C}(X,Y))$ for all $X,Y$ in $\text K(\mathcal C)$.