Let X be a projective, connected and smooth scheme defined over an algebraically closed field k. In this paper we prove that a tower of finite torsors (i.e., under the action of finite k-group schemes) can be dominated by a single finite torsor. Let G be any finite k-group scheme and Y any G–torsor over X pointed in y ∈ Y (k); we define over Y , which may not be reduced, in a very natural way the categories of Nori-semistable and essentially finite vector bundles. These categories are proved to be Tannakian. Their Galois k-group schemes π S (Y, y) and π N (Y, y), respectively, thus generalize the S–fundamental and the Nori fundamental group schemes. The latter still classifies all the finite torsors over Y , pointed over y. We also prove that they fit in short exact sequences involving π S (X, x) and π N (X, x) respectively, where x is the image of y.