  The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal
surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which
contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to
conjugation. In all cases except the cyclic quotient singularities the
connected component $G_1$ of the unit equals ${\Bbb C}^*$. The induced action
of $G$ on the minimal good resolution of $X$ embeds the finite group $G/G_1$
into the automorphism group of the central curve $E_0$ of the exceptional
divisor. We describe $G/G_1$ as a subgroup of ${\rm Aut}\: E_0$ in case $E_0$
is rational as well as for simple elliptic singularities. Moreover, sufficient
conditions for $G$ to be a direct product $G_1 \times G/G_1$ are presented.
Finally, it is shown that $G/G_1$ acts faithfully on the integral homology of
the link of $X$.
