I need the name of this group to explain some differences in various notions of the group of a topological fiber bundle. I would think that for each element in the group of a bundle, there is a loop in the base space for the bundle that causes that transformation on the fiber space. Most (all?) definitions seem to allow more group elements than this.

Perhaps it should be called the **symmetry group** of the topology.
The reals: [0, 1] have a topological symmetry group of 2.
The isomorphism x → x^{2} is equivalent to the identity.
The isomorphism x → 1−x is not.

It seems clear to me that the group for the Möbius strip, as a bundle, should be the two group.

The automorphisms of a topological space, form a topological group. The connected components thereof form a group like the group we seek.

Perhaps the group should be called the topology’s discrete symmetry group.