Not quite, although if an observer were to move at the group speed would see a time independent intensity distribution although there would still be time variations within the pulse. Maxwell's equations in a medium are not Lorentz covariant, simply because there is material medium. To understand this, imagine an extreme case where the the wave propagates through a stratified medium; the refractive indices will be time varying for an observer moving normal to the layers. Or, from another standpoint: imagine the medium as a lattice of atoms. As you move relative to it, the Lorentz Fitzgerald contraction means that you see a lattice with its period in the direction of motion shrunken, whereas the lattice periods normal to the motion are unchanged. Put simply, an isotropic material becomes anisotropic, as the material's optical density in the direction of motion is increased.
A peculiar aspect to this problem is that group velocities add according to the wonted relativistic 3-velocity addition law, but phase velocities do not; there is a more complicated transformation for the latter. If you could magically color the pulse so that different surfaces of constant phase were given their own different colors, you'd see a stationary pulse with colors moving though it. However, even if group and phase speeds are the same (nondispersive medium), if one could see microscopically, one could see charges shuttling backward and forth so that the magnetic fields are not the simple magnetostatic fields that we would get if we imagined the charges in the medium to be moving by. There are electric and magnetic fields somewhat like what one would see if one could see the fields in a resonant cavity.
These conclusions are fairly messy to reach, but the basic line of reasoning is that the wave's phase field must be a Lorentz scalar. From this postulate, one works as described in:
Kirk T. McDonald, "Index of Refraction of a Moving Medium"
to conclude the results above.