Exceptional groups and the s-arc-transitivity of vertex-primitive digraphs, II

In this paper, we study the primitive actions of almost simple groups with socle $E_7(q)$ or $E_8(q)$ on an $s$-arc-transitive digraph. Our motivation goes back to the question of whether $s$ is bounded above for finite connected $G$-vertex-primitive $s$-arc-transitive digraphs that are not directed cycles. The question has been reduced by Giudici and Xia to the case where $G$ is almost simple. This work succeeds our 2025 paper (Yin and Chen), which addressed ${}^3\text{\hspace{0.17em}⁣}{D}_4(q)$, $G_2(q)$, ${}^2\text{\hspace{0.17em}⁣}{F}_4(q{)}^{\mathrm{\prime }}$, $F_4(q)$, $E_6(q)$, and ${}^2\text{\hspace{0.17em}⁣}{E}_6(q)$. Together with Chen, Giudici, and Praeger’s work on ${}^2\text{\hspace{0.17em}⁣}{B}_2(q)$ and ${}^2\text{\hspace{0.17em}⁣}{G}_2(q)$, it answers the question for all exceptional groups.