Proof of Inflation-Restriction Exact Sequence by Spectral Sequence

Question

It is well know that the inflation-restriction exact sequence $$0 \rightarrow H^1(G/H , M^H) \rightarrow H^1(G,M)\rightarrow H^1(H,M)$$ arises naturally from the Hochschild-Serre exact sequence. While this is a Grothendieck spectral sequence $$R^pF_1R^qF_2M \Rightarrow R^{p+q}(F_1F_2)M$$ with $F_1 = (-)^{G/H}$ and $F_2 = (-)^H$. However, $H^1(H,M)$ is a derived functor of $(-)^H$ in $Mod_H$ instead of $Mod_G$. Then how can the proof by the spectral sequence make sense?