We conduct numerical simulations to map out the phase diagram and critical behavior of a lattice Higgs model composed of two massless staggered fermion fields forming a doublet under a global SU (2) and coupled to a scalar field in the adjoint representation of the group. The scalar action consists of a potential comprising quadratic and quartic terms and a scalar kinetic term. At fixed quartic coupling we explore a two-dimensional parameter space finding a massless symmetric phase at weak coupling and a massive symmetric phase (SMG phase) at strong coupling. An intermediate anti-ferromagnetic phase separates these two regimes. These results are consistent with leading order weak and strong coupling expansions. We find that the critical lines bounding the intermediate phase merge at a unique point where all fermion bilinear condensates vanish but fermion susceptibilities diverge as non-trivial powers of the lattice size. We conjecture that this merged point corresponds to a multicritical point and may describe a phase consisting of a condensate of certain topological defects.