  The fixed point spectra of Morava E-theory $E_n$ under the action of finite
subgroups of the Morava stabilizer group $\mathbb{G}_n$ and their K(n)-local
Spanier--Whitehead duals can be used to approximate the K(n)-local sphere in
certain cases. For any finite subgroup F of the height 2 Morava stabilizer
group at p=2 we prove that the K(2)-local Spanier--Whitehead dual of the
spectrum $E_2^{hF}$ is $\Sigma^{44}E_2^{hF}$. These results are analogous to
the known results at height 2 and p=3. The main computational tool we use is
the topological duality resolution spectral sequence for the spectrum
$E_2^{h\mathbb{S}_2^1}$ at p=2.
