Given a permutation group G and a subgroup U, the ring of invariants of U is larger than the one for G. Thus, one can ask for a U-invariant that is not G-invariant. This is exactly the problem one has to solve for the computation of Galois groups.

As we deal with permutation groups, the first try is to use block systems to attack the problem. However in some cases this does not yield suitable invariants. I will explain how the transfer map and the Reynolds operator can be used to construct invariants. In the case of intransitive groups we will use the main theorem of subdirect products to solve the problem.

The methods suffice to construct invariants for all examples arising from the transitive permutation group database of all transitive groups up to degree 32.