The embedding approach to the electronic structure of local perturbations in extended systems is based on the fundamental assumption that beyond a certain region around the defect, the properties of the environment are not altered by the presence of the defect. In many computational schemes the resulting subdivision of the defect system into a central and an external region is defined in terms of orbital basis functions. The fundamental embedding assumption then translates into a partitioning of matrix representations, accompanied by fixing the external region contributions to their values in the unperturbed reference system. With the help of density functional cluster-in-cluster embedding calculations we have investigated the quality of this assumption without introducing any additional approximation as usually done to arrive at a computationally feasible embedding scheme. The fundamental embedding assumption is found to cause spurious virtual orbital admixtures to the density matrix which lead to artifacts in the results of embedding calculations. To minimize these undesirable effects, a special "class orthogonalization" scheme has been employed. It allows a perfect reproduction of the defect induced charge density changes as judged by cluster-in-cluster model calculations for a hydrogen substitutional defect in large Li-n clusters (with n up to 309). However, equilibrium geometries, total energies, and vibrational frequencies calculated with this embedding scheme do not exhibit any improvement over results from calculations employing the corresponding nonembedded model clusters. The reason for this failure which prevents the expected converence of the calculated results with increasing cluster size is analyzed. Thus, from a pragmatic point of view, "naked" cluster models are preferable, at least for metal substrates, due to their relative computational simplicity. Possible techniques to either avoid the virtual orbital admixtures or to improve the quality of the total energies obtained from the embedding calculations are discussed together with the drawbacks of these schemes.