  A few approximate schemes to solve the Hedin equations self-consistently
introduced in (Phys. Rev. B 94, 155101 (2016)) are explored and tested for the
3D electron gas at metallic densities. We calculate one electron spectra,
dielectric properties, compressibility, and correlation energy. Considerable
reduction in the calculated band width (as compared to the self consistent GW
result) has been found when vertex correction was used for both polarizability
and self energy. Generally, it is advantageous to obtain the diagrammatic
representation of polarizability from the definition of this quantity as a
functional derivative of the electronic density with respect to the total field
(external plus induced). For self energy, the first order vertex correction
seems to be sufficient for the range of densities considered. Whenever it is
possible, we compare the accuracy of our vertex-corrected schemes with the
accuracy of the self-consistent quasi-particle GW approximation (QSGW), which
is less expensive computationally. We show that QSGW approach performs poorly
and we relate this poor performance with an inaccurate description of the
screening in the QSGW method (with an error comprising a factor 2-3 in the
physically important range of momenta).
