\(\displaystyle \psi:N\rightarrow G\)

where \(\displaystyle G\) is a abelian group, how do I extend this homomorphism to the whole of \(\displaystyle G(N)\) where \(\displaystyle G(N)\) is the envelopping Gothendieck group of \(\displaystyle N\) to obtian the homomorphism

\(\displaystyle \tilde{\psi}:G(N)\rightarrow G\)