There is a connection between permutation groups and permutation patterns: for any subgroup G of the symmetric group Sℓ and for any n ≥ ℓ, the set of n-permutations involving only members of G as ℓ-patterns is a subgroup of Sn. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels as automorphism groups of relations.