We look at the problem of when the map on algebraic fundamental groups induced by inclusion of the regular locus into a normal variety is an isomorphism. For each point of the variety, we identify a local obstruction, governed by the local algebraic fundamental group of the point. Finally, we show the equivalence of these obstructions being finite and the existence of a Galois etale cover of the regular locus where the map on fundamental groups is an isomorphism.