Partial actions on quotient spaces and globalization

Given a partial action of a topological group $G$ on a space $X$, we determine properties $\mathcal{P}$ which can be extended from $X$ to its globalization. We treat the cases when $\mathcal{P}$ 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 $G\mathrm{/}H$ on the orbit space $X\mathrm{/}\text{\hspace{0.17em}⁣}\sim ,$ applications to invariant metrics and inverse limits are presented.