From Matthew D. Shwartz Quantum Field Theory textbook, he writes:
"Most of the time, a symmetry of a classical theory is also a symmetry of the quantum theory based on the same Lagrangian. When it is not, the symmetry is said to be anomalous."
A priori, how is this even possible? If a symmetry is present in a classical Lagrangian, and the quantum theory is based on the same Lagrangian, and the Lagrangian completely specifies the dynamics, then wouldn't the symmetry being present in the Lagrangian necessitate the symmetry is in the quantum theory?