  In this paper we present results of numerical simulations and some
(analytical) approximations of a compact $U(1)\ltimes\ZZ_2$ lattice gauge
theory, including an extra bare mass term for Alice fluxes. The subtle
interplay between Alice fluxes and (Cheshire) magnetic charges is analysed. We
determine the phase diagram and some characteristics of the model in three and
four dimensions. The results of the numerical simulations in various regimes,
compare well with some analytic approximations.
