  As a spin analog of the plethystic Murnaghan-Nakayama rule for Schur
functions, the plethystic Murnaghan-Nakayama rule for Schur $Q$-functions is
established with the help of the vertex operator realization. This generalizes
both the Murnaghan-Nakayama rule and the Pieri rule for Schur $Q$-functions. A
plethystic Murnaghan-Nakayama rule for Hall-Littlewood functions is also
investigated.
