An unusual example of a universal automorphism group

Abstract

Let $M$ be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We refer to the class of structures embeddable in $M$ as the $\ddot{I}‰$-age of $M$. We consider the following two properties of $M$: we say that $M$ has a universal automorphism group if, for each $A$ in the $\ddot{I}‰$-age of $M$, there is an embedding $\text{Aut}(A)\to \text{Aut}(M)$, and we say that $M$ has group-extensible $\ddot{I}‰$-age if, for each $A$ in the $\ddot{I}‰$-age of $M$, there is an embedding $A\to M$ such that each automorphism of the image extends to an automorphism of $M$ and the extension map preserves group composition. It is immediate that if $M$ has group-extensible $\ddot{I}‰$-age, then $M$ has a universal automorphism group. We give an example of a Fraïssé structure with a universal automorphism group whose $\ddot{I}‰$-age is not group-extensible, showing that the above two properties are not equivalent.