  We compute the Picard group of the category of $K(2)$-local module spectra
over the ring spectrum $E^{hC_4}$, where $E$ is a height 2 Morava $E$-theory
and $C_4$ is a subgroup of the associated Morava stabilizer group. This group
can be identified with the Picard group of $K(2)$-local $E$-modules in genuine
$C_4$-spectra. We show that in addition to a cyclic subgroup of order 32
generated by $ E\wedge S^1$ the Picard group contains a subgroup of order 2
generated by $E\wedge S^{7+\sigma}$, where $\sigma$ is the sign representation
of the group $C_4$. In the process, we completely compute the $RO(C_4)$-graded
Mackey functor homotopy fixed point spectral sequence for the $C_4$-spectrum
$E$.
