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We give strengthened versions of the Herwig-Lascar and Hodkinson-Ottoextension theorems for partial automorphisms of finite structures. Suchstrengthenings yield several combinatorial and group-theoretic consequences forhomogeneous structures. For instance, we establish a coherent form of theextension property for partial automorphisms for certain Fraisse classes. Wededuce from these results that the isometry group of the rational Urysohnspace, the automorphism group of the Fraisse limit of any Fraisse class that isthe class of all $\mathcal{F}$-free structures (in the Herwig--Lascar sense),and the automorphism group of any free homogeneous structure over a finiterelational language, all contain a dense locally finite subgroup. We also showthat any free homogeneous structure admits ample generics.

Coherent extension of partial automorphisms, free amalgamation, and automorphism groups