\'Etale Covers and Local Algebraic Fundamental Groups

Let $X$ be a normal noetherian scheme and $Z\subseteq X$ a closed subset of codimension $\ge 2$. We consider here the local obstructions to the map ${\widehat{\pi }}_1(X\mathrm{\backslash }Z)\to {\widehat{\pi }}_1(X)$ being an isomorphism. Assuming $X$ has a regular alteration, we prove the equivalence of the obstructions being finite and the existence of a Galois quasi-'{e}tale cover of $X$, where the corresponding map on fundamental groups is an isomorphism.