  Let $(X,\bar x)$ be a pointed connected noetherian scheme. In this note, we
give characterizations for the vanishing of the second \'etale homotopy group
$\pi^{\rm \'et}_2(X,\bar x)$ in terms of splitting profinite-\'etale covers of
$X$, and by means of universal covering spaces of the Artin-Mazur-Friedlander
\'etale homotopy type $Et(X)$. In particular, this provides certain classes of
schemes for which the Brauer map is surjective.
