Let $X$ be a nice complex variety and fix a basepoint $x_0$. Denote $\pi:=\pi_1(X,x_0)$. Let $\mathbb{V}$ be a local system on $X$, and consider the cohomology groups $H^i(\pi, \mathbb{V}_{x_0}), H^i(X,\mathbb{V})$. Clearly, they are isomorphic for $i=0$.
I heard that there is a Leray spectral sequence that computes the group cohomologies using the singular cohomologies, and that it gives a map from the $i$'th singular ones to the $i'th$ group ones. Does anyone know how to construct this map? When is it injective?