We investigate the possibility of using transient absorption of a coherent bound electron wave packet in hydrogen as an attosecond pulse characterization technique. In a recent work, we have shown that photoionization of such a coherent bound electron wave packet opens up for pulse characterization with unprecedented temporal accuracy-independent of the atomic structure-with maximal photoemission at all kinetic energies given a wave packet with zero relative phase (Pabst and Dahlstrom Phys. Rev. A 94 13411 (2016)). Here, we perform numerical propagation of the time-dependent Schrodinger equation and analytical calculations based on perturbation theory to show that the energy-resolved maximal absorption of photons from the attosecond pulse does not uniquely occur at a zero relative phase of the initial wave packet. Instead, maximal absorption occurs at different relative wave packet phases, distributed as a non-monotonous function with a smooth -pi/2 shift across the central photon energy (given a Fourier-limited Gaussian pulse). Similar results are also found in helium. Our finding is surprising, because it implies that the energy-resolved photoelectrons are not mapped one-to-one with the energy-resolved absorbed photons of the attosecond pulse.