Can a symmetry of a subset of the plane always be extended to the entire plane?

Question

Given some compact subset of the plane and a symmetry of that subset, i.e. a bijective isometry from that set to itself, it seems that in any example I come up with the symmetry can be written as some combination of translations, rotations, and reflections - in other words it may be extended as a symmetry of the plane. This leads me to conjecture that this is always the case. Indeed, this MathOverflow question seems to cite a stronger result, but I cannot see what the referenced material has to do with this fact.