  We show that for a partial action $\eta$ with closed domain of a compact
group $G$ on a profinite space $X$ the space of orbits $X/\!\sim_G$ is
profinite, this leads to the fact that when $G$ is profinite then the
enveloping space $X_G$ is also profinite. Moreover, we provide conditions for
the induced quotient map $\pi_G: X\to X/\!\sim_G$ of $\eta$ to have a
continuous section. To prove this result we use enveloping actions to construct
partial actions of quotient groups on orbit spaces. Relations with continuous
sections of the quotient map induced by the enveloping action of $\eta$ are
also considered. At the end of this work we treat with Exel's semigroup and the
action groupoid, in particular we give conditions for them to be profinite
spaces.
