The paraxial propagation of a relativistic electron beam in a solid target is examined, within a three-dimensional model of particles interacting with the target electron return current via a diffusive electromagnetic field. Simulations of a modulated beam show amplification of the modulation seed, with growth rates comparing reasonably well with the linear analysis of the model. Scenarios of beam fragmentation are observed and discussed in more realistic conditions, when beam collisions on both target ions and electrons and the resulting solid heating and ionization are taken into account.