The energetic electron content in the Van Allen radiation belts surrounding the Earth can vary dramatically at several timescales, and these strong electron fluxes present a hazard for spacecraft traversing the belts. The belt response to solar wind driving is, however, largely unpredictable, and the direct response to specific large-scale heliospheric structures has not been considered previously. We investigate the immediate response of electron fluxes in the outer belt that are driven by sheath regions preceding interplanetary coronal mass ejections and the associated wave activity in the inner magnetosphere. We consider the events recorded from 2012 to 2018 in the Van Allen Probes era to utilise the energy- and radial-distance-resolved electron flux observations of the twin spacecraft mission. We perform a statistical study of the events by using the superposed epoch analysis in which the sheaths are superposed separately from the ejecta and resampled to the same average duration. Our results show that the wave power of ultra-low frequency Pc5 and electromagnetic ion cyclotron waves, as measured by a Geostationary Operational Environmental Satellite (GOES), is higher during the sheath than during the ejecta. However, the level of chorus wave power, as measured by the Van Allen Probes, remains approximately the same due to similar substorm activity during the sheath and ejecta. Electron flux enhancements are common at low energies (<1 MeV) throughout the outer belt (L = 3-6), whereas depletion predominantly occurs at high energies for high radial distances (L > 4). It is distinctive that the depletion extends to lower energies at larger distances. We suggest that this L-shell and energy-dependent depletion results from the magnetopause shadowing that dominates the losses at large distances, while the wave-particle interactions dominate closer to the Earth. We also show that non-geoeffective sheaths cause significant changes in the outer belt electron fluxes.