Let $X$ be compact connected. Let $F$ be the skyscraper sheaf over $p$ for a group $G$. Let $H^1(X,F)$ be the first Cech cohomology of the sheaf. We want to show $\ker(d_1) = im(d_0)$ is trivial. It suffices to show $d_0(C_0) = 0$ Choose some cover of $X$, if $p \in U_{i}$, then for $g \in F(U_i)$ we have $g|_{U_i \cap U_j} - g|_{U_i \cap U_j} = 0$.
This seems too easy, am I missing something?