Algorithm to test a graph for $t$-transitivity
If you had an oracle from Graph Isomorphism, then for constant $t$ you can check whether a graph is $t$-transitive. For example, to check if a graph is $1$-transitive, for every vertex $i$ create a variant $G_i$ of your original graph by attaching an "appendage" at vertex $i$, say a very long path or a very large star. The graph is transitive if all $G_i,G_j$ are isomorphic.