  Let $G$ be a group acting $2$-transitively on the boundary of a locally
finite tree, and exclude the situation (which is a genuine exception) where $G$
has both $\mathrm{P}\Gamma\mathrm{L}_3(4)$ and
$\mathrm{P}\Gamma\mathrm{L}_3(5)$ as local actions. We show that for each
half-tree $T_a$, the local action of the rigid stabilizer of $T_a$ at the root
of $T_a$ contains the soluble residual of the point stabilizer of the local
action of $G$. In particular, $G$ is locally prosoluble if and only if its
local actions have soluble point stabilizers; if $G$ is not locally prosoluble,
then it has micro-supported action on the boundary. We also prove some strong
restrictions on the local actions of end stabilizers in $G$. These results are
partly inspired by Radu's classification of groups acting
boundary-$2$-transitively on trees with local action containing the alternating
group, and partly based on the author's recent classification of finite
permutation groups that preserve an equivalence relation, act faithfully on
blocks and act transitively on pairs of points from different blocks.
