Let M be an ultrahomogeneous relational structure (eg the random graph, or Q as a linear order). Let A be a substructure of M. If A is finite, then all automorphisms of A extend to automorphisms of M, immediately by the definition of ultrahomogeneity. What happens if A is infinite? We will discuss several examples and a general theorem for free amalgamation classes. This is ongoing work with Alessandro Codenotti, Aristotelis Panagiotopoulos and Jeroen Winkel.