Is graph automorphism Karp-reducible to graph isomorphism under hidden subgroup representation?
I think your focus on the rigid case of GI limits you too much. Instead phrase (non-rigid) GI as an HSP in the same way, but now the goal is to determine the size of the hidden subgroup, or a generating set. The difference between the isomorphic and non-isomorphic cases will be a factor of 2 in the order of the hidden subgroup. If you phrase the problem as finding generators of the hidden subgroup, then the question is just whether any generator switches $\Gamma_1$ and $\Gamma_2$.

Now, an instance of "HSP-GA" corresponding to a graph $\Gamma$ is given by the function from $S_n \to M_n(\mathbb{C})$ defined by $f(\pi) = A(\pi(\Gamma))$ where $A(\cdot)$ denotes the adjacency matrix. In particular, $f(1) = A(\Gamma)$. Then apply the usual Karp reduction from GA to GI to get a pair of graphs $\Gamma_1, \Gamma_2$, and consider the HSP-GI instance, of the type described in the preceding paragraph, corresponding to the pair $\Gamma_1, \Gamma_2$ (that is, the disjoint union $\Gamma_1 \cup \Gamma_2$).