Hom-killer

Definition with symbols
Let $$A$$ and $$B$$ be groups. A Hom-killer for $$A$$ with respect to $$B$$ is a group $$C$$ containing $$A$$ such that $$A$$ lies in the kernel of any homomorphism from $$C$$ to $$B$$.

Relation with potential characteristicity
if $$C$$ is a Hom-killer for $$A$$ with respect to $$B$$, then $$A$$ is relatively characteristic with respect to $$A \times B$$ inside $$C \times B$$.

This shows that if a Hom-killer exists for $$A$$ with respect to $$B$$ exists then $$A$$ is a potentially relatively characteristic subgroup of $$A \times B$$. Note that, in fact, since potentially relatively characteristic equals normal this does not provide any additional information.