This problem was reported by Lei Chen.

Lemma 2.37(ii) is not true if $\mathfrak G$ is a digraph. Using the notation of the book, a $G$-vertex-transitive digraph $\mathfrak G$ is $(G,2)$-arc-transitive if and only if $G_\alpha$ factorises as $G_\alpha=G_{\alpha, \alpha_1} G_{\alpha,\alpha_2}$ for a 2-arc $(\alpha_1, \alpha, \alpha_2)$ (see for instance Lemma 2.1 in [GX]).

This part of the lemma is only used for undirected graphs in the rest of the book and so this causes no problem in subsequent arguments.

[GX] Michael Giudici and Binzhou Xia.ย Vertex-quasiprimitive 2-arc-transitive digraphs.ย Ars Mathematica Contemporanea, 14 (2018), 67โ82.