Variants of the de Jong fundamental group

For a rigid space X, we answer two questions of de~Jong about the category CovᵃdmX of coverings which are locally in the admissible topology on X the disjoint union of finite \'etale coverings: we show that this class is different from the one used by de~Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of Cov\'etX (with the analogous definition) correspond precisely to locally constant sheaves for the pro-\'etale topology defined by Scholze.