The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at T=∞ and magnetization M=0 , an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization M0 in one of the configurations upon quenching the system at T C, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent θ D=1.915(3) , which is much larger than the exponent θ=0.197 characteristic of the initial increase of the magnetization M(t). Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic (R2(t)) grows with an exponent z∗ ≈ η ≈ 1.9, which is the same, within error bars, as the exponent θ D. However, the survival probability of the epidemics reaches a plateau so that δ=0. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at T D ≃ 0.51TC, where all the measured observables exhibit power laws with exponents θ D=1.026(3), δ=0.133(1), and z∗=1.74(3).