Classifying bundles with homogeneous space as fibers A principal $G$-bundle over a space $X$ is classified by the homotopy classes of maps $[X,BG]$, where $BG$ is the classifying space of the group $G$. My question is what can we do about this when the fiber is a homogeneous space.
For a general fiber bundle $F\hookrightarrow E\rightarrow X$, the classification is usually done by $[X,B\rm{Diff}(F)]$ and is obviously a difficult beast. But for a homogeneous space $G/H$, for $G$ a Lie group and $H$ a closed subgroup, can we say something about the space $B\rm{Diff}(G/H)$? Is there any way to relate them to say $BG$ and $BH$?
More specifically what I'm looking for is a way to classify Lagrangian Grassmann bundles over a certain manifold. It is known that the fiber $LG(n,2n)=U(n)/O(n)$ is a homogeneous space. So is there any classification result in this direction?
Any help is appreciated!
 A: Unfortunately, just because $G/H$ a homogeneous space doesn't mean we get much control over $B\text{Diff}(G/H)$. At most you get something nice if $G/H$ is dimension $\leq 3$. 
What you might have better luck with is by thinking of the Lagrangian Grassmannian as a homogeneous space: it has an effective action of $U(n)$, which means an injective homomorphism $f: U(n) \to \text{Diff}(U(n)/O(n))$. 
You could restrict your transition functions to lie in the image of $f$. (One says you've "reduced the structure group to $U(n)$".) In this case, classifying Lagrangian Grassmannian bundles with structure group $U(n)$ is the same thing as classifying principal $U(n)$-bundles over your space: given a principal $U(n)$-bundle $P$ over a base $B$ (so it has a free right action of $U(n)$ whose quotient is $B$), we may take $P \times_{U(n)} \left(U(n)/O(n)\right)$ to get a Lagrangian Grassmannian bundle over $B$. 
(Alternatively, look at the transition functions: they're exactly the same for the two bundles, the fibers are just different!)
So you've then reduced yourself to calculating $[X, BU(n)]$, aka, finding the set of rank $n$ complex vector bundles over your space.
