  We investigate the properties of a four-dimensional conformal field theory
possessing a fermionic higher-spin current $Q_{\alpha(2k) \dot{\alpha}}$. Using
a computational approach, we examine the number of independent tensor
structures contained in the three-point correlation functions of two fermionic
higher-spin currents with the conserved vector current $V_{m}$, and with the
energy-momentum tensor $T_{m n}$. In particular, the $k = 1$ case corresponds
to a "supersymmetry-like" current, that is, a fermionic conserved current with
identical properties to the supersymmetry current which appears in $\mathcal{N}
= 1$ superconformal field theories. However, we show that in general, the
three-point correlation functions $\langle Q Q T\rangle $, $\langle \bar{Q} Q
V\rangle $ and $\langle \bar{Q} Q T\rangle $ are not consistent with
$\mathcal{N}=1$ supersymmetry
