  Let $N$ be a nilpotent group normal in a group $G$. Suppose that $G$ acts
transitively upon the points of a finite non-Desarguesian projective plane
$\mathcal{P}$. We prove that, if $\mathcal{P}$ has square order, then $N$ must
act semi-regularly on $\mathcal{P}$.
  In addition we prove that if a finite non-Desarguesian projective plane
$\mathcal{P}$ admits more than one nilpotent group which is regular on the
points of $\mathcal{P}$ then $\mathcal{P}$ has non-square order and the
automorphism group of $\mathcal{P}$ has odd order.
