Countable structures with a fixed group of automorphisms

We prove that, given a countable group G, the set of countable structures (for a suitable language L)U_G whose automorphism group is isomorphic to G is a complete coanalytic set and if G ≄ H then U_G is Borel inseparable from U_H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures for L whose automorphism group is isomorphic to ℤ_p^ℕ ,p a prime number, is Π^1_1&Σ^1_1-complete