We show that the author's notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed $G$-categories, recently introduced for the purposes of 3-manifold topology [31]. The Galois extensions $C \rtimes S$ are studied in detail, and we determine for which $g$ in $G$ non-trivial objects of grade $g$ exist in $C \rtimes S$.