Lemma 5

If $c$ is a 2-coboundary, then so is $\widetilde{c}_s$.

Let $t: H\rightarrow F$ be a mapping such that $c=\partial t$. Let $\widetilde{t}_s: G\rightarrow F$ be the mapping defined by

for $g\in G$. It suffices to prove that $\widetilde{c}_s=\partial \widetilde{t}_s$, which follows from the relations $\widetilde{c}_s(g_1, g_2) =\prod_{x\in H\backslash G}^{s(x)^{-1}s(x)g_1s(x\cdot g_1)^{-1}}t(s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})$

for $g_1, g_2\in G$.