The additive group of the integers is the set of isomorphisms of a line with evenly spaced points and a direction.
Alternatively, take a directed graph with:
V = The integers
E = {x, y | x - y = 1}
and that graph has the same isomorphism group.
Replace the nodes of that graph with asymmetric graphs, and the resultant undirected graph again has the same isomorphism group.
Well, then you are missing the entire (additive) group of integers... But you are right, in that automorphisms are the most important example, and they also play a huge theoretical role (as representations of abstract groups). This is what gives the group theory its importance in math and physics.