Q:

Homomorphisms of Topological Groups which are Automatically Fiber Bundles? 

Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way.  When is this a fiber bundle over G? ( It will then be a K-principal bundle over G).   
I'm writing a paper where I make a claim about when this holds.  I thought I had a reference but when I went looking for it, my claim was not in the reference.  
I don't want to consider examples that are too pathological so lets assume everything is Hausdorff and paracompact. ( However if people are familiar with the more general setting, I'd be curious about that too!). 
Clearly a necessary condition is that $G \cong E/K$, so let's assume this is the case.  By homogeneity it is enough to show that f admits a local section in a neighborhood of the identity element of G. So my question is equivalent to asking if there are conditions I can impose on E and G which will ensure that f admit local sections near the identity of G. 
I know by work of G. Segal ("Cohomology of Topological Groups" Symposia Math.  Vol IV 1970 pg 377, in the appendix) that if G is abelian and locally contractible then the sequence $$G \to EG \to BG$$
is of this kind.  
I want to know: 

Does this hold when K is locally contractible? 
What if K is globally contractible? 
Are there any simple (but not tautological) conditions I can put on K, E, or G to make this hold? 

I'd also like to know some reasonable examples where this fails to be a principal bundle (if there are any).  

A:

The following statement follows from results of Palais (see theorem 2.3.3 in "On the existence of slices for actions of non-compact Lie groups").  When $G$ is a Lie group, any principal $G$-bundle (in the sense of Husemoller's book, which does not assume local triviality) whose total space is completely regular is actually locally trivial. 
It follows that when $H$ is a Lie group, and a closed subgroup of a Hausdorff topological group $G$, then $G\to G/H$ is a locally trivial principal $H$-bundle. 
In a slightly different direction, it is a theorem of Skljarenko (theorem 15 of "On the topological structure of locally bicompact groups and their quotient spaces") that when $G$ is a locally compact Hausdorff group, and $H$ is a closed subgroup of $G$, then $G\to G/H$ is a Hurewicz fibration.  In particular, if $G/H$ is (relatively) locally contractible then $G\to G/H$ is a locally trivial principal $H$-bundle.  Again, by a result of Skljarenko, it is also enough to assume that $G/H$ is finite dimensional to conclude local triviality.  See also Mostert's "Sections in principal fibre spaces".