Recently, a class of many-body Hamiltonians have been found to show exact periodic dynamics in the continuum of the energy spectrum where thermalization and chaos is expected. This phenomenon is known as the (exact) quantum many-body scar (QMBS) dynamics. We show that a wide range of QMBS models, plus many new ones, can be unified and constructed following one prescription based on a non-Abelian symmetry that we call the quasisymmetry. In addition we show that the recently discovered asymptotic scar states in the XY-model exist in generic QMBS models as the Nambu-Goldstone modes of the corresponding quasisymmetry.