  We construct super Yang-Mills theories with ${\cal N}=2, 4$ supersymmetries
on the two-dimensional square lattice keeping one or two supercharges exactly.
Along the same line as the previous paper \cite{sugino}, the construction is
based on topological field theory formulation. In order to resolve the problem
of degenerate classical vacua encountered in the previous paper, we consider
two kinds of modifications of the action. For one of them, any supersymmetry
breaking terms do not need to be introduced, and the formulations exactly
realize some of supersymmetries at the lattice level. Our lattice actions flow
to the desired continuum theories without any fine-tuning of parameters.
