Two questions (more details below):


1. Let $G$ be a compact Lie group and $X$ a $G$-space such that all stabilizer
subgroups are conjugate to a fixed $H \leq G$. Denote by $\pi: X
\to X/G$ the quotient map. Under which conditions on $X$ is $\pi$ a Serre fibration?
2. Let $G$ be as above and $F$ a *free* $G$-space. Under which conditions
is the canonical map $F\_{hG} \to F/G$ from the homotopy quotient to the
quotient a weak equivalence?


All spaces are assumed to be CGWH spaces.




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**Relation between 1. and 2.:**


I am most interested in 2.. However, if $F$ is a free $G$-space, $EG$ denotes
Bar construction, such that $(F \times EG)/G$ models $F\_{hG}$, and both $F$ and
$F \times EG$ are as in 1., i.e., the respective quotient maps are Serre
fibrations, then it is not hard to see that $F\_{hG} \to F/G$ is a weak
equivalence, using the long exact sequence of homotopy groups.


So, any condition for 1. that holds for $EG$ and is stable under products
provides a condition for 2.




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**Results known to me:**


* A sufficient condition for 1. is being a completely regular Hausdorff space
(also known as a Tychonoff space) by a result in Bredon's "Introduction to
Compact Transformation groups." He shows that the quotient map is a fiber
bundle in this case. However, the Tychonoff property does not seem to be
preserved by CGWH products, so it will not hold for many constructions which
one would like to have results as in 1. or 2. for.
* A sufficient condition for 2. should be that $F$ is a (retract of a) free
$G$-CW-complex by using model category arguments. However, this also a rather
severe condition that can be hard or impossible to check in practice.


Thus, I would like to know if there any other known results, preferably with a
reference, that improve the sufficient conditions outlined above. Any results for more arbitrary
topological groups will also be appreciated.



When is $X_{hG} \to X/G$ a weak equivalence for $X$ a free $G$-space, $G$ compact Lie?
Assertions 1. and 2. hold for spaces that are (compactly generated and) Hausdorff (we still have to require $G$ to be compact Lie). This is Theorems A.10 and A.8 of my recent preprint <https://arxiv.org/abs/1612.04267v3>. An application, where the "results known to me" from above are not sufficient, can be found there as well.


(It's hard to find out what happens for spaces that are CGWH but not Hausdorff. This is related to the following question: [Compact Lie group action on non-Hausdorff (but CGWH) space with Hausdorff quotient](https://mathoverflow.net/questions/255629/))


