The development of a modulated (2,2) antiphase ordering in the two-dimensional anisotropic next-nearest-neighbor Ising model is studied following a quench from the disordered phase to a low-temperature unstable state. The domain growth and the dynamical structure factor are found to evolve anisotropically. The structure factor is shown to satisfy scaling for a variety of choices of a length scale. The growth laws for different definitions of length scale are shown to be most consistent with an effective power-law exponent of n0.50. The dynamical roles of different types of domain walls and vertices present in this model are discussed, with an emphasis placed on the need for theoretical studies of this model.