I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise. 
Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for eg. a curve) as well as a short exact sequence of smooth connected groups
$1\to K\to G\stackrel{\pi}{\to} H\to 1$ 
(if one prefers one can work aver $\mathbb{C}$ and take the groups to be connected complex Lie groups). Since we're dealing with principal bundles pick your favourite topology, say $\tau$ (fpqc, etale, classical analytic - I hope that the answer will not depend in an essential way on this).


Now let us also fix $F\_H$ an $H$-bundle on $X$. I'm interested in understading the category $\mathcal{K}\_{F\_H}$ of liftings of $F\_H$ to $G$ (it is a stackover $X$, right?). 
This is defined as follows:


for an $X$-scheme $f:T\to X$ we have:


objects: pairs $(E,\alpha)$ where $E$ is a $G$-bundle on $T$ and $\alpha:\pi\_\*(E)\to f^\*(F\_H)$ an isomorphism of $H$-bundles.


morphisms: from a pair $(E,\alpha)$ to $(E',\alpha')$ are isomorphisms of $G$-bundles $u:E\to E'$ that are compatible with $\alpha$ and $\alpha'$, i.e. $\alpha' \pi\_\*(u) = \alpha$.


In case $K=\ker(\pi)$ is abelian $H$ acts naturally by conjugation on $K$ and one can form the following group scheme over $X$: $K\_{F\_H} = F\_H\stackrel{H}{\times}K$.


One can see rather easily (for e.g. using local charts) that $\mathcal{K}\_{F\_H}$ is a gerbe over $X$ which is, locally on $X$, isomorphic to $BK\_{F\_H}\times X$.


The obstruction of $F\_H$ to have a lift to a $G$-bundle is an element $\xi\in H^2\_{\tau}(X,K\_{F\_H})$. Moreover, if $\xi=0$ the gerbe is trivial and hence the stack $\mathcal{K}\_{F\_H}$ is equivalent to the stack of $K\_{F\_H}$-bundles on $X$.


I hope I haven't messed things up yet.


My question is: what if $K$ is not abelian? then there's no natural action of $H$ on $K$ (even if the sequence is split the conjugation action depends on the splitting). Where lives the obstruction of lifting an $H$-bundle? And, what interests me more actually, what is the structure of the category of liftings of $F\_H$ to $G$ once one knows it has a lift?


I'm pretty sure the answer should be in Giraud's book somewhere but I find it quite hard to read so I'm having troubles detecting the right place to look. Any references or comments would be helpful. 



liftings of principal bundles
The broad outlines of how this business works don't depend on the fact that you're working with varieties so let me work with spaces instead, by which I mean homotopy types. 


Let $f : X \to BG$ be a principal bundle and let $BH \to BG$ be a map along which you'd like to lift. Then the space of lifts of $f$ to $BH$ (up to homotopy) is precisely the space of (homotopy) sections of the pullback $X \times\_{BG} BH$, or equivalently the space of sections of the $G/H$-bundle associated to $f$. (Here by $G/H$ I mean the homotopy fiber of $BH \to BG$, which may or may not be the actual quotient $G/H$, whatever that means in your setting.)


The nicest case is when $BH \to BG$ is itself a homotopy fiber, or equivalently when it fits into a fiber sequence


$$\Omega Y \to BH \to BG \to Y$$


for some space $Y$ (which must necessarily deloop $G/H$; in this case the associated $G/H$-bundle is a "principal $\Omega Y$-bundle"). Then the space of lifts of $f$ to $BH$ is, by the universal property of the homotopy fiber, precisely the space of nullhomotopies of the composite map $X \to BG \to Y$. In the very nicest cases $Y$ is an Eilenberg-MacLane space.


In particular, if $G, H$ fit into a short exact sequence


$$K \to H \to G$$


then $BK, BH, BG$ fit into a fiber sequence


$$BK \to BH \to BG.$$


If $K$ is not only abelian but central in $H$, then this fiber sequence deloops to a fiber sequence


$$BK \to BH \to BG \to B^2 K$$


and so we can take $Y = B^2 K$, in which case the existence of a nullhomotopy of the composite $X \to BG \to Y$ is equivalent to the corresponding cohomology class in $H^2(X, K)$ vanishing. If $K$ is abelian but not central then this cohomology group needs to be taken with local coefficients. But sometimes $Y$ is more complicated than this. For example, if $G = \text{Spin}$ and $H = \text{String}$ then $Y = B^4 \mathbb{Z}$ and the existence of a lift is controlled by a cohomology class $\frac{p\_1}{2} \in H^4(X, \mathbb{Z})$. 


In general, though, $G/H$ just isn't a loop space, and so can't be delooped, which means that $Y$ doesn't exist and the obstruction theory gets harder. The obstructions are a sequence of classes in cohomology $H^{k+1}(X, \pi\_k(G/H))$ with local coefficients, each of which is only well-defined provided that the previous one vanishes. A typical case here is $G = O(2n), H = U(n)$. 


