Is there examples of non-isomorphic cospectral graphs having


1. Non-isomorphic automorphism groups?
2. Isomorphic automorphism groups?



On cospectral graphs
Yes, there are lots of such examples of strongly regular graphs. E.g. [Chang graphs](https://en.m.wikipedia.org/wiki/Chang_graphs) give an example of for 1).


For 2), many cospectral graphs have trivial automorphism groups (thus isomorphic as groups).


Or, for a concrete example 2), take point graphs of [generalised quadrangles](https://en.m.wikipedia.org/wiki/Generalized_quadrangle) GQ(3,3). These are two non-isomorphic cospectral strongly regular graphs on 40 vertices.


