  We study the conductance $g$ of an electron interferometer created in a two
dimensional electron gas between a nanostructured contact and the depletion
region induced by the charged tip of a scanning gate microscope. Using
non-interacting models, we study the beating pattern of interference fringes
exhibited by the images giving $g$ as a function of the tip position when a
parallel magnetic field is applied. The analytical solution of a simplified
model allows us to distinguish between two cases: (i) If the field is applied
everywhere, the beating of Fabry-P\'erot oscillations of opposite spins gives
rise to interference rings which can be observed at low temperatures when the
contact is open between spin-split transmission resonances. (ii) If the field
acts only upon the contact, the interference rings cannot be observed at low
temperatures, but only at temperatures of the order of the Zeeman energy. For a
contact made of two sites in series, a model often used for describing an
inversion-symmetric double-dot setup, a pseudo-spin degeneracy is broken by the
inter-dot coupling and a similar beating effect can be observed without
magnetic field at temperatures of the order of the interdot coupling.
Eventually, numerical studies of a quantum point contact with quantized
conductance plateaus confirm that a parallel magnetic field applied everywhere
or only upon the contact gives rises to similar beating effects between
spin-split channel openings.
