It is easy to extend group-2-cocycles to smash-products with Nichols algebras over the group (just trivially). The same certainly doesn't work for nontrivial liftings.
As I would like to check a construction of mine also over the latter, I wonder how to proceed...a first step is surely to treat liftings as Doi twists (by Masuoka for small $U_q$'s, by other papers for other cases, see my respective MA-Question)
- I tried explicitely conjugating the trivial extended group-2-cocycle with the Doi twist, but the calculations seemed to lead to nowhere :-(
- Does it help to abstractly consider the category equivalence induced by the Doi twist? Intuitively I would expect the Galois objects to be already determined by this structure? But how do I get the right explicit expression?
Thanks in advance :-)