The Simon's problem involves a function which takes binary strings as inputs. One seeks to find the period of the function which acts on those inputs. In the standard method, the first register has the capability to encode the inputs.
The graph isomorphism problem is defined over two input graphs. The hidden subgroup representation of the problem involves a function which doesn't take the graphs as inputs. Rather it takes permutation over the vertices as input. Those permutations are agnostic of the input graphs. In the standard method, the first register doesn't have the capability to encode the inputs i.e. the graphs.
Can anyone explain why the HSP representations of these two problem are so different in the sense that in the standard method of an hidden subgroup approach the first register of one of them encodes the inputs but the other doesn't?