Reconstructing structures with the strong small index property up to bi-definability

Let K be the class of countable structures M with the strong small index property and locally finite algebraicity, and K∗The class of M ∈ K such that aclM(fag) = fag for every a 2 M. For homogeneous M ∈ K, we introduce what we call the expanded group of automorphisms of M, and show that it is second-order definable in Aut(M). We use this to prove that for M;N ∈ K∗, Aut(M) and Aut(N) are isomorphic as abstract groups if and only if (Aut(M);M) and (Aut(N);N) are isomorphic as permutation groups. In particular, we deduce that for N0-categorical structures the combination of the strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's (1994) well-known 89-interpretation technique. Finally, we show that every finite group can be realized as the outer automorphism group of Aut(M) for some countable N0-categorical homogeneous structure M with the strong small index property and no algebraicity.