Automorphisms and strongly invariant relations

We investigate characterizations of the Galois connection Aut - sInv between sets of finitary relations on a base set A and their automorphisms. In particular, for A= ω₁ , we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under sInv Aut . Our structure (A, R) has an ω -categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.