Singular varieties and infinitesimal non-commutative Witt vectors

Given a projective variety $X$ over an algebraically closed field $k$, M. V. Nori introduced in 1976 a group scheme $\pi(X)$ which accounts for principal bundles $P\to X$ with finite structure, obtaining in this way an amplification the etale fundamental group. One drawback of this theory is that it is quite difficult to arrive at an explicit description of $\pi(X)$, whenever it does not vanish altogether. To wit, there are no known non-trivial examples in the literature where $\pi(X)$ is local, or local of some given height, etc. In this paper we obtain a description of $\pi(X)$ through amalgamated products of certain non-commutative local group schemes - we called them infinitesimal non-commutative Witt group schemes - in the case where $X$ is a non-normal variety obtained by pinching a simply connected one.