  We consider ${\mathcal N}=1$ superconformal field theories in four dimensions
possessing an additional conserved spinor current multiplet $S_{\alpha}$ and
study three-point functions involving such an operator. A conserved spinor
current multiplet naturally exists in superconformal theories with ${\mathcal
N}=2$ supersymmetry and contains the current of the second supersymmetry.
However, we do not assume ${\mathcal N}=2$ supersymmetry. We show that the
three-point function of two spinor current multiplets and the ${\mathcal N}=1$
supercurrent depends on three independent tensor structures and, in general, is
not contained in the three-point function of the ${\mathcal N}=2$ supercurrent.
It then follows, based on symmetry considerations only, that the existence of
one more Grassmann odd current multiplet in ${\mathcal N}=1$ superconformal
field theory does not necessarily imply ${\mathcal N}=2$ superconformal
symmetry.
