This paper presents the derivation, analysis, and benchmarking of the fully dynamic G3W2 self-energy for finite systems. The G3W2 self-energy incorporates all possible time orderings of three Green's functions (G) and two dynamic screened Coulomb interactions (W). The analytic formula and its imaginary frequency counterpart are derived, with the latter enabling the treatment of larger molecules. The G3W2 self-energy is evaluated on established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. It is shown that neither the G3W2 self-energy nor any of the investigated approximations improve over the one-shot $ G_0W_0 $ method. However, quasi-particle self-consistent GW HOMO energies are slightly improved by the addition of the G3W2 self-energy correction. The self-consistent update of the screened Coulomb interaction leads to an overall sign change of the vertex correction to the frontier quasiparticle energies. The paper also introduces a new approximation, 2SOSEX, which better approximates the complete G3W2 self-energy. The results show that the G3W2 self-energy does not improve over the simpler GW self-energy for most systems, but it provides a more accurate description of core levels. The paper concludes that the G3W2 self-energy does not improve over the simpler GW self-energy for most systems, but it provides a more accurate description of core levels. The paper also suggests that future work should focus on combining vertex corrections in both the self-energy and the screened Coulomb interaction.This paper presents the derivation, analysis, and benchmarking of the fully dynamic G3W2 self-energy for finite systems. The G3W2 self-energy incorporates all possible time orderings of three Green's functions (G) and two dynamic screened Coulomb interactions (W). The analytic formula and its imaginary frequency counterpart are derived, with the latter enabling the treatment of larger molecules. The G3W2 self-energy is evaluated on established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. It is shown that neither the G3W2 self-energy nor any of the investigated approximations improve over the one-shot $ G_0W_0 $ method. However, quasi-particle self-consistent GW HOMO energies are slightly improved by the addition of the G3W2 self-energy correction. The self-consistent update of the screened Coulomb interaction leads to an overall sign change of the vertex correction to the frontier quasiparticle energies. The paper also introduces a new approximation, 2SOSEX, which better approximates the complete G3W2 self-energy. The results show that the G3W2 self-energy does not improve over the simpler GW self-energy for most systems, but it provides a more accurate description of core levels. The paper concludes that the G3W2 self-energy does not improve over the simpler GW self-energy for most systems, but it provides a more accurate description of core levels. The paper also suggests that future work should focus on combining vertex corrections in both the self-energy and the screened Coulomb interaction.