Odd Steinhaus problem for finite sets

MathOverflow Asked by domotorp on November 30, 2020

Call a finite subset $S$ of the plane with an even number of points an odd Jackson set, if there is an $Asubset mathbb R^2$ such that $A$ meets every congruent copy of $S$ in an odd number of points.

Are there any odd Jackson sets?

For a bit of history of related questions (and an explanation of the nomenclature), see here.
For my motivation, see here.

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