The Strong Small Index Property for Free Homogeneous Structures

We show that in algebraically locally finite countable homogeneous structures with a free stationary independence relation the small index property implies the strong small index property. We use this and the main result of [15] to deduce that countable free homogeneous structures in a locally finite relational language have the strong small index property. We also exhibit new continuum sized classes of ${\mathrm{\aleph }}_0$-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.