For a fixed positive integer n, if we colour the edges of the complete graph on a countable vertex set randomly with n colours, there is a particular configuration which arises with probability 1. Its group G of strict (colour-preserving) automorphisms is simple, and the outer automorphism group of G is Sn (corresponding to permutations of the colours). The automorphism group of G splits over G if and only if n is odd. If n is even and not a multiple of 8, there is a finite supplement to G; this is unknown in the remaining case, where it reduces to a problem about finite groups.
In the case n = 2, the configuration consists of the celebrated random graph and its complement. I will also talk about embeddings of its automorphism group as a group of automorphisms of a filter, or homeomorphisms of a topology, on the vertex set.