The following proof of the inflation-restriction exact sequence is taken from Milne's notes on class field theory. My question is: why does $\phi':G/H \to M$ actually take values in $M^H$?
In other words, why is $$h\phi(g)-hgm_0+hm_0=\phi(g)-gm_0+m_0 $$ where $\phi:G \to M$ is a crossed homomorphism, meaning $\phi(g_1g_2)=g_1\phi(g_2)+\phi(g_1)$. Also, we are assuming for $h\in H$, $\phi(h)=hm_o-m_0$.
The key is that $H$ is normal in $G$. Since $\phi'$ is a crossed homoorphism and $\phi'(h)=0$ for $h \in H$, we have $$\phi'(hg)=h\phi'(g)+\phi'(h)=h \phi'(g)$$
and $$\phi'(gh)=g\phi'(h)+\phi'(g)=\phi'(g)$$ The second displayed equation says $\phi'$ is constant on left cosets of $H$. Since $H$ is normal, left cosets and right cosets are the same, so $\phi'$ is constant on right cosets. Since $hg$ and $g$ are in the same right coset $Hg$, we have $\phi'(hg)=\phi'(g)$. Combining this with the first displayed equation gives $h\phi'(g)=\phi'(g)$, i.e $\phi'(g) \in M^H$.