Here is a question closely related to a characterization of standard atoms (not to be confused with standard alphabets:). It asks whether a bound on the number of automorphisms is equivalent to a bound on the numer of color-preserving automorphisms, in every Fraisse class of finite relational structures.
For a Fraisse class of finite structures over a finite relational vocabulary, are the following two conditions equivalent? Below, by a colored structure from we mean a structure from together with a mapping from its carrier to some finite set of colors.
A color class is the set of elements mapped to the same color.
1. The structures in have boundedly many automorphisms: for some number , every structure in has at most automorphisms.
2. The colored structures in have boundedly many automorphisms: for every number there is a number such that every colored structure in with color classes of size bounded by , has at most color-preserving automorphisms.