On iterated universal extensions and Nori's fundamental group of nilpotent bundles

Let $k$ be a field of characteristic $0$, $X$ be a geometrically connected, smooth and proper variety over $k$ and $x\in X(k)$ be a base point. Using the notion of iterated universal extensions, we show that Nori’s fundamental group ${\pi }_1^N(X,x)$ of nilpotent bundles is uniquely determined by the coherent cohomology groups ${\mathrm{H}}^i(X)={\mathrm{H}}^i(X,{\mathcal{O}}_X)$, $i=1,2$, and the cup product $\cup :{\mathrm{H}}^1(X)\otimes {\mathrm{H}}^1(X)\to {\mathrm{H}}^2(X)$. This can be seen as an analogue of a classical fact on the de Rham fundamental group of compact Kähler manifolds. As a byproduct, we also determine low degree group cohomology of the trivial representation of ${\pi }_1^N(X,x)$, notably in degree $2$.