(I imagine you know this, but for the sake of future readers) I think the word Torsor is relevant here :) https://math.ucr.edu/home/baez/torsors.html is a nice informal introduction.
(I imagine you know this, but for the sake of future readers) I think the word Torsor is relevant here :) https://math.ucr.edu/home/baez/torsors.html is a nice informal introduction.