  We determine the Gross-Hopkins duals of certain higher real $K$-theory
spectra. More specifically, let $p$ be an odd prime, and consider the Morava
$E$-theory spectrum of height $n=p-1$. It is known, in the expert circles, that
for certain finite subgroups $G$ of the Morava stabilizer group, the homotopy
fixed point spectra $E_n^{hG}$ are Gross-Hopkins self-dual up to a shift. In
this paper, we determine the shift for those finite subgroups $G$ which contain
$p$-torsion. This generalizes previous results for $n=2$ and $p=3$.
