  We study numerically the SU(2) Kazakov-Migdal model of `induced QCD'. In
contrast to our earlier work on the subject we have chosen here {\it not} to
integrate out the gauge fields but to keep them in the Monte Carlo simulation.
This allows us to measure observables associated with the gauge fields and
thereby address the problem of the local $Z_2$ symmetry present in the model.
We confirm our previous result that the model has a line of first order phase
transitions terminating in a critical point. The adjoint plaquette has a clear
discontinuity across the phase transition, whereas the plaquette in the
fundamental representation is always zero in accordance with Elitzur's theorem.
The density of small $Z_2$ monopoles shows very little variation and is always
large. We also find that the model has extra local U(1) symmetries which do not
exist in the case of the standard adjoint theory. As a result, we are able to
show that two of the angles parameterizing the gauge field completely decouple
from the theory and the continuum limit defined around the critical point can
therefore not be `QCD'.
