  We investigate stabilizers of finite sets of rational points in Cantor space
for the Higman-Thompson groups $V_{n,r}$. We prove that the pointwise
stabilizer is an iterated ascending HNN extension of $V_{n,q}$ for any $q\geq
1$. We also prove that the commutator subgroup of the pointwise stabilizer is
simple, and we compute the abelianization. Finally, for each $n$ we classify
such pointwise stabilizers up to isomorphism.
