Suppose you have a map of topological spaces $X\to S$, an $S$-group $G\to S$ (i.e. a group object in $\mathrm{Top}_{/S}$), an action of $G$ on $X$ relative to $S$ which is free and properly discontinuous.
QUESTION. Under which possibly additional conditions is it true that
a) there is a well-defined topological quotient $X/^{S}G\to S$ with a $G$-invariant structure map $X\to X/^{S}G$ over $S$ and the universal property that any map $X\to Y$ over $S$ which is $G$-invariant factors through $X/^{S}G$?
b) you can lift paths or even homotopies along $X\to X/^{S}G$?
About point (1), I guess it should be true in good generality, but I would still like to have a clear result in mind.
E.g. when $S=*$ and $G$ is a compact Lie group point (2) should be true. I am interested in the case when $S$ is nontrivial, but any hypotheses about the group are ok.
