  Let $k$ be an algebraically closed field of positive characteristic $p>0$ and
$C \to {\mathbb P}^1_k$ a $p$-cyclic cover of the projective line ramified in
exactly one point. We are interested in the $p$-part of the full automorphism
group $Aut_k C$. First we prove that these groups are exactly the extra-special
$p$-groups and groups G which are subgroups of an extra-special group E such
that $Z(E) \subseteq G$. The paper also describes an efficient algorithm to
compute the $p$-part of $\Aut_k C$ starting from an Artin-Schreier equation for
the cover $C \to {\mathbb P}^1_k$.
  The interest for these objects initially came from the study of the stable
reduction of $p$-cyclic covers over the $p$-adics. There the covers $C \to
{\mathbb P}^1_k$ naturally arise and their automorphism groups play a major
role in understanding the arithmetic monodromy. Our methods rely on previous
work by Stichtenoth whose approach we have adopted.
