2-cocycles of a given Hopf algebras $H$ no longer form a group, but a groupoid between different Doi twists of the Hopf algebra $H,L$. The subgroup of "lazy" 2-cocycles precisely preserve the underlying Hopf algebra. They are usually presented as $H$-$L$- resp. $H$-$H$-Bigalois Objects.

Now we know from Schauenburg, that grouprings $H$ by cocommutativity have only lazy 2-cocycles (those of the group) and $Bigal(H)=Aut(G)\ltimes H^2(G,k^*)$

We also know the structure of lazy 2-cocycles (and resp. Bigalois objects) on tensor products (Schauenburg) and Drinfel'd doubles (Bichon,Carnovale) in terms of the smaller Hop algebra one's and the pairings between them....

...but NOT the respective larger entire Bigalois grouppoid....

...but I want FAR LESS: namely just the Bigalois grouppoid on the Drinfel'd double of a group! I searched the web, but could note even find a description of the Doi twists...

Any idea or hint would be greatly appreciated! Thanx in advance