  We produce for each natural number $n \geq 3$ two 1--parameter families of
Riemann surfaces admitting automorphism groups with two cyclic subgroups
$H_{1}$ and $H_{2}$ of orden $2^{n}$, that are conjugate in the group of
orientation--preserving homeomorphism of the corresponding Riemann surfaces,
but not conjugate in the group of conformal automorphisms.
  This property implies that the subvariety $\mathcal{M}_{g}(H_{1})$ of the
moduli space $\mathcal{M}_{g}$ consisting of the points representing the
Riemann surfaces of genus $g$ admitting a group of automorphisms topologically
conjugate to $H_{1} $ (equivalently to $H_{2}$) is not a normal subvariety.
