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We carry out a numerical investigation of three-dimensional linear perturbations in a self-similar ablation wave in slab symmetry, representative of the shock transit phase of a pellet implosion in in-ertial confinement fusion (ICF). The physics of ablation is modeled by the equation of gas dynamics with a nonlinear heat conduction as an approximation for radiation transport. Linear perturbation responses of the flow, its external surface and shock-wave front, when excited by external pressure or heat-flux perturbation pulses, are computed by fully taking into account the flow compressibility, non-uniformity and unsteadiness. These responses show the effective propagation, at supersonic speeds, of perturbations from the flow external surface through the whole conduction region of the ablation wave, beyond its Chapman-Jouguet point, and up to the ablation front, after the birth of the ablation wave. This supersonic forward propagation of perturbations is evidenced by means of a set of appropriate pseudo-characteristic variables and is analyzed to be associated to the 'heat-conductivity' waves previously identified in [Clarisse et al., J. Fluid Mech. 848, 219-255 (2018)]. Such heat-conductivity linear waves are found to prevail over heat diffusion as a feedthrough mechanism [Aglitskiy et al., Phil. Trans. R. Soc. 368, 1739-1768 (2010)] for perturbations of longitudinal characteristic lengths of the order of-or larger than-the conduction region size, and long transverse wavelengths with respect to this region size, and over time scales shorter to much shorter than the shock transit phase duration. This mechanism which results from the dependency of the heat conductivity on temperature and density in conjunction with a flow temperature stratification, is expected to occur for other types of nonlinear heat conductions-e.g. electron heat conduction-as well as to be efficient at transmitting large scale perturbations from the surrounding of an ICF pellet to its inner compressed core at later times of its implosion. Besides, the proposed set of pseudo-characteristic variables are recommended for analyzing perturbation dynamics in an ab-lation flow as it furnishes additional propagation information over the fundamental linear modes of fluid dynamics [Kovásznay, J. Aero. Sci. 20(10), 657-674 (1953)], especially in the flow conduction region.