The investigation herein is two-fold: specialization to a homogeneous obstacle of recent work (Lambert M et al 1998 Inverse Problems 14 1265-83) carried on the retrieval of an inhomogeneous cylindrical obstacle buried in a half-space in the transverse electric (or H) polarization case; extension from the usual case of complete wavefield data (known amplitude and phase) to the more severe case of amplitude-only data (absent phase). The developed inversion method belongs to the class of modified gradient methods. The field distribution (here, the magnetic field) and the distribution of the obstacle parameters (here, the permittivity and conductivity) are simultaneously sought in a search domain. This is done by minimizing a two-component objective function, one of which is characterizing the satisfaction of the wave equations, the other the data fit. But now the electrical parameters of the sought obstacle are prescribed beforehand; this allows one to equate an appropriate complex-valued contrast function either to 0 (outside the obstacle) or to a known constant (inside). Two variants of a binary-constrained modified gradient algorithm are developed accordingly, tailored to either complete or phaseless data. Numerical experimentation illustrates how they behave in a variety of obstacle configurations, for both exact and erroneous prescribed contrasts.