= Solution
The axial $U(1)$ is anomalous, while vector fermion number survives. In a left-handed convention, put $\widetilde\psi_R=(\psi_R)^c$. The continuous quantum symmetry and charges are
$$
SU(N_f)_L\times SU(N_f)_R\times U(1)_V,
$$
$$
\begin{array}{c|c|c|c|c}
&SU(2)_{\rm gauge}&SU(N_f)_L&SU(N_f)_R&U(1)_V\\\hline
\psi_L&\mathbf2&\mathbf{N_f}&\mathbf1&+1\\
\widetilde\psi_R&\mathbf2&\mathbf1&\overline{\mathbf{N_f}}&-1
\end{array}
$$
up to finite quotients and the surviving discrete axial symmetry.
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