When does a morphism $f:X \to C$ from a group extension $X \twoheadrightarrow G$ factor through $G$?

Question

All the letters denote abelian groups (all finite except for $C$). Let $Y$ be an extension of $G$ by $B$ and let $f: Y \to C$ be a group homomorphism. Now, I want to find the maximal $G$-subextension $X \subseteq Y$ by $A \subseteq B$, s.t. there is a map $\tilde{f}:G \to C$ with $f\circ \iota = \tilde{f} \circ \pi_X$, where $\pi_X:X \to G$ is the projection of $X$ to $G$ and $\iota$ denotes the inclusion of $X$ in $Y$. Is this a known problem? If so, where can I find something about it? Of course, I am basically interested in existence and uniqueness statements..