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 $\mathbf{2}$, 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.