  This work extends the results of [Garde and Hyv\"onen, Math. Comp.
91:1925-1953] on series reversion for Calder\'on's problem to the case of
realistic electrode measurements, with both the internal admittivity of the
investigated body and the contact admittivity at the electrode-object
interfaces treated as unknowns. The forward operator, sending the internal and
contact admittivities to the linear electrode current-to-potential map, is
first proven to be analytic. A reversion of the corresponding Taylor series
yields a family of numerical methods of different orders for solving the
inverse problem of electrical impedance tomography, with the possibility to
employ different parametrizations for the unknown internal and boundary
admittivities. The functionality and convergence of the methods is established
only if the employed finite-dimensional parametrization of the unknowns allows
the Fr\'echet derivative of the forward map to be injective, but we also
heuristically extend the methods to more general settings by resorting to
regularization motivated by Bayesian inversion. The performance of this
regularized approach is tested via three-dimensional numerical examples based
on simulated data. The effect of modeling errors related to electrode shapes
and contact admittances is a focal point of the numerical studies.
