Residually finite groups with finitely many automorphism orbits are locally finite of finite exponent; finitely generated locally graded ones are finite; and Mal'cev Q-completions of free nilpotent groups have this property only when r=2 and c=3 or c≤2.
Osin,Small cancellations over relatively hyperbolic groups and embedding theorems, Annals of Mathematics, 172 (2010) 1-39
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Some families of locally graded groups with finitely many orbits under automorphisms
Residually finite groups with finitely many automorphism orbits are locally finite of finite exponent; finitely generated locally graded ones are finite; and Mal'cev Q-completions of free nilpotent groups have this property only when r=2 and c=3 or c≤2.