  We consider the inhomogeneous fifth-order nonlinear Schr\"{o}dinger (ifoNLS)
equation with nonzero boundary condition in detailed. Firstly, the spectral
analysis of the scattering problem is carried out. A Riemann surface and affine
parameters are first introduced to transform the original spectral parameter to
a new parameter in order to avoid the multi-valued problem. Based on Lax pair
of the ifoNLS equation, the Jost functions are obtained, and their analytical,
asymptotic, symmetric properties, as well as the corresponding properties of
the scattering matrix are established systematically. For the inverse
scattering problem, we discuss the cases that the scattering coefficients have
simple zeros and double zeros, respectively, and we further derive their
corresponding exact solutions. Moreover, some interesting phenomena are found
when we choose some appropriate parameters for these exact solutions, which is
helpful to study the propagation behavior of these solutions.
