Characterization of the 4-canonical birationality of algebraic 3-folds, II

For nonsingular projective 3-folds $X$ (of general type) whose geometric genus $p_g=h^0(X, K_X)$ is $\geq 5$, the birationality of the fourth canonical map $\varphi_{4,X}=\Phi_{|4K_X|}$ was characterized by D.-Q. Zhang and the first author in 2008. This paper aims at characterizing the birationality of $\varphi_{4,X}$ for those $X$ with $p_g=4$.