We consider the PermCode problem to decide, given a representation of a permutation group G and a parameter k, whether there is a non-trivial element of G with support at most k. This problem generalizes several problems in the literature. We introduce a new method that allows to reduce the maximal orbit size of the group being considered while maintaining elements with small support in the group. Using this technique we provide upper and lower bounds for several variants of the parameterized Hypergraph Isomorphism Problem that extend previous results on parameterized Graph Isomorphism.