  We study the phase structure of a $Z_2$ lattice Higgs-Yukawa system with
different forms of chirally invariant lattice fermion actions: naive, two types
of non-local actions, and a mirror fermion action. The calculations are
performed in the variational mean field approximation with contribution of
fermionic determinant being calculated in a ladder approximation. We
demonstrate that the phase structure of the system crucially depends on the
form of the lattice fermion action, despite the fact that the Higgs field
couples to fermions in the same local way. A possibility of defining chiral
gauge theories based on mirror fermion action is briefly discussed.
