This paper presents a comprehensive analysis and benchmarking of the fully dynamic G3W2 self-energy for finite systems. The G3W2 self-energy, which includes all possible time orderings of three Green's functions and two dynamically screened Coulomb interactions, is derived and analyzed. The analytic formula for G3W2 and its complex frequency counterpart are presented, allowing for the treatment of larger molecules. The accuracy of G3W2 is evaluated using well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. The authors also explore the relationship between G3W2 and simpler static approximations, such as SOSEX (static screened second-order exchange), and propose a more consistent approximation named 2SOSEX. However, the results show that neither G3W2 nor any of its approximations significantly improve over one-shot G0W0 with a good starting point. Only the quasi-particle self-consistent G0W0 HOMO energies are slightly improved by the addition of the G3W2 correction, which is attributed to the self-consistent update of the screened Coulomb interaction leading to a sign change in the vertex correction for frontier quasiparticle energies. The paper concludes with a discussion on the limitations of the current approach and suggestions for future research.This paper presents a comprehensive analysis and benchmarking of the fully dynamic G3W2 self-energy for finite systems. The G3W2 self-energy, which includes all possible time orderings of three Green's functions and two dynamically screened Coulomb interactions, is derived and analyzed. The analytic formula for G3W2 and its complex frequency counterpart are presented, allowing for the treatment of larger molecules. The accuracy of G3W2 is evaluated using well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. The authors also explore the relationship between G3W2 and simpler static approximations, such as SOSEX (static screened second-order exchange), and propose a more consistent approximation named 2SOSEX. However, the results show that neither G3W2 nor any of its approximations significantly improve over one-shot G0W0 with a good starting point. Only the quasi-particle self-consistent G0W0 HOMO energies are slightly improved by the addition of the G3W2 correction, which is attributed to the self-consistent update of the screened Coulomb interaction leading to a sign change in the vertex correction for frontier quasiparticle energies. The paper concludes with a discussion on the limitations of the current approach and suggestions for future research.