Partial actions on quotient spaces and globalization

Given a partial action of a topological group G on a space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable, and having invariant metric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space of X; applications to invariant metrics and inverse limits are presented.