A theory describing the extended oscillations in secondary-electron (se) spectra and their connections with the short-range atomic structure in solids is developed. We investigate the two main mechanisms for the occurrence of SEs: direct knocking out of SEs from the core level by primary electrons, and a second-order process, called autoionization, going through excitation of a core-level electron into some intermediate states followed by emission of the final SEs. Interference Of a direct wave and waves reflected from neighbouring atoms are considered both in the final and in the intermediate states. We derive the cross sections for SEs created in single crystals and polycrystals. In the latter case the signal is described by the two EXAFS-like oscillating functions which are controlled by the final and intermediate energies. The intermediate wave diffraction is described by the EXAFS-like backscattering in all cases. The extended fine structure is expressed, in an explicit form, in terms of coordinates and the scattering characteristics of neighbouring atoms, which opens up new possibilities for obtaining information on the short-range atomic structure near surfaces.