factorisation of a trivial homomorphism

Question

let $G$ be an abelian group. and $f:G\rightarrow \{0\}$ be the trivial homomorphism. suppose there exists $G\stackrel{g}{\rightarrow} H \stackrel{h}{\rightarrow} \{0\}$ such that $f=h\circ g$ does this imply that necessarely $H=\{0\}$ if not then under what condition this is true?

Answer 1

No, it does not. Take $H=G$, $g=id$, and $h$ the trivial homomorphism.