I think the group of the bundle must be a subgroup of the homotopy group of the base.
I think there can be ‘the group’ of an equivalence class of a fiber bundles, but not a particular fiber bundle.
Two FBs are equivalent if their base and fiber are respectively homeomorphic and they have equivalent atlases.
Two atlases are equivalent if there is a bijection between the respective charts and …
Groups of Particular Bundles