I have found several definitions of the group associated with a fiber bundle. There is much variation in detail. In each case they are groups of transformations on the fiber space. I think that all are ambiguous in allowing groups that are larger than intended. Perhaps ‘topological fiber bundle’ is to be distinguished from ‘fiber bundle’. If so I mean ‘topological fiber bundle’ here.

Here are some specific bundles and the ‘intended’ group as given in examples drawn from several sources:

Any Cartesian product
The group with one element
The Möbius strip
The group with two elements
Steenrod: “The Topology of Fibre Bundles” 1951
Steenrod finds it necessary to introduce coordinate bundles in order to define the bundle’s group. This is like an ‘atlas’ with many ‘charts’, one for each member of an open cover of the base space. A loop in the base space will traverse these charts and each time it crosses from one to another, a transformation on the fiber space occurs. Upon reaching the origin of the loop the total transformation on the fiber must be an element of the group.
Penrose
Wikipedia
If we take the stance that the 3 spaces, bundle, base & fiber, together with projection determine the fiber bundle and thereby determine the group, at least up to isomorphism, then we may have to specify a concept of homomorphism for fiber bundles.

I would propose that the three spaces be homeomorphic and the projection and isomorphism commute.

Steenrod gives the normal torus as the Cartesian produce of two circles. He also gives, in presumed distinction, the ‘twisted torus’ as follows: The base and fiber space are both circles, the bundle space is their Cartesian product and the projection takes <a, b> to b but the two ends of the ‘tube’ are rotated by 180°. I would have said that they were homomorphic if that word had been defined.
Is there a topological way to say that the closed unit interval has two sorts of map onto itself, one sort flips the ends and the other sort does not? In this particular case the line segment is defined in terms of the reals which have a total ordering such that {x | a < x < b} is open for all a and b. Any continuous bijection from the open segment to itself is monotonic either increasing or decreasing. This distinction does not generalize well. I want a point set distinction. The n-ball likewise has two equivalence classes of continuous permutations. This trick works only for n=1.