  We develop a method for describing the Galois action on the superspecial
locus of the Siegel moduli space in characteristic $p$. Using this description,
we give a modern treatment for the results of Ibukiyama and Katsura [Compos.
Math., 1994] concerning the $\mathbb{F}_p$-rational points and the trace of a
Hecke operator of Atkin-Lehner type. This also leads to analogues with
level-$N$ structure. The trace of the Hecke operator can be reduced into one
term (instead of finitely many terms a priori) by the simple trace formula when
$N$ is large.
