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We calculate the beating of $h/2e$ and $h/e$ periodic oscillations of theflux-dependent critical supercurrent $I_c(\Phi)$ through a quantum spin-Hallinsulator between two superconducting electrodes. A conducting pathway alongthe superconductor connects the helical edge channels via a non-helicalchannel, allowing an electron incident on the superconductor along one edge tobe Andreev reflected along the opposite edge. In the limit of small Andreevreflection probability the resulting even-odd effect is described by$I_c\propto|\cos(e\Phi/\hbar)+f|$, with $|f|\ll 1$ proportional to theprobability for phase-coherent inter-edge transmission. Because the sign of $f$depends on microscopic details, a sample-dependent inversion of the alternationof large and small peaks is a distinctive feature of the beating mechanism forthe even-odd effect.

Even-odd flux quanta effect in the Fraunhofer oscillations of an edge-channel Josephson junction