On the subsystems of certain sofic shifts

For an aperiodic subshift of finite type Y and for a subshift X with topological entropy less than the topological entropy of Y, a theorem is proved in Krieger: On the subsystems of topological Markov chains, Ergodic Theory & dynamical systems 1982 2, 195-202, that says that the necessary condition on the periodic points of X and Y for the existence of an embedding of X into Y is also sufficient for the existence of an embedding of X into Y. In this note we point out that this theorem extends to certain classes of sofic shifts as target shifts.