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.