  We study the Josephson effect in a quantum spin Hall system coupled to a
localized magnetic impurity. As a consequence of the fermion parity anomaly,
the spin of the combined system of impurity and spin-Hall edge alternates
between half-integer and integer values when the superconducting phase
difference across the junction advances by $2\pi$. This leads to characteristic
differences in the splittings of the spin multiplets by exchange coupling and
single-ion anisotropy at phase differences, for which time-reserval symmetry is
preserved. We discuss the resulting $8\pi$-periodic (or $\mathbb{Z}_4$)
fractional Josephson effect in the context of recent experiments.
