A group homomorphism $\phi:G'\to G$ gives rise to $\Bbb{Z}$-module homomorphism $\tilde{\phi}:\Bbb{Z}[G']\to\Bbb{Z}[G'']$ defined as $\tilde{\phi}(\sum\limits_{h\in G'} a_h h)=\sum\limits_{h\in G} a_h \phi(h)$. Give conditions under which the sequence $0\to\Bbb{Z}[G']\to\Bbb{Z}[G]\to\Bbb{Z}[G'']$ is short exact for a given sequence $G'\to G\to G''$?
Now we are given a exact sequence $G' \xrightarrow{\phi}G\xrightarrow{\psi}G''$. I can prove that if $\psi$ is injective then $\tilde{\phi}$ is injective and if $\psi$ is surjective then $\tilde{\psi}$ is surjective. I can also prove that $\tilde{\psi}\circ \tilde{\phi}=0$ i.e. $\text{Image}(\tilde{\phi})\subseteq\text{Ker}(\tilde{\psi})$. But I'm unable to prove $\text{Image}(\tilde{\phi})=\text{Ker}(\tilde{\psi})$. Do I need any more condition on the sequence $G'\to G\to G''$?