We give an overview of some recent developments around the topic of the small index property for automorphism groups of countable structures. Specifically, we overview the following results of the authors:
(1) Countable free homogeneous structures in a locally finite irreflexive relational language have the strong small index property, thus generalizing Cameron's analogous result for the random graph.
(2) For $\aleph_0$-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's well-known $\forall \exists$-interpretation technique.
(3) The automorphism group of Hall's universal locally finite group has the strong small index property and it is complete.