Consider two fiber bundles with structure group $G$, which are isomorphic as fiber bundles without structure group (i.e., with the full group of homeomorphisms of the fiber as structure group). Are they also isomorphic as fiber bundles with structure group $G$?
In other words, the structure group constrains the transition functions of a fiber bundle, so the map from structure-group-$G$ bundles to no-structure-group bundles is not surjective. Does the structure group also constrain possible equivalences between transition functions such that the map is also not injective?
Edit: Maybe I should rather ask: Is there a natural notion of isomorphism between fiber bundles where the structure-group enters non-trivially?