  We take a direct approach to computing the orbits for the action of the
automorphism group $\mathbb{G}_2$ of the Honda formal group law of height $2$
on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{\mathbb{G}_2}
\cong \mathbb{F}_p$. The result is new for $p=2$ and $p=3$. For primes $p\geq
5$, the result is a consequence of computations of Shimomura and Yabe and has
been reproduced by Kohlhaase using different methods.
