  \textit{Dessins d'enfants} (hypermaps) are useful to describe algebraic
properties of the Riemann surfaces they are embedded in. In general, it is not
easy to describe algebraic properties of the surface of the embedding starting
from the combinatorial properties of an embedded dessin. However, this task
becomes easier if the dessin has a large automorphism group.
  In this paper we consider a special type of dessins, so-called \textit{Wada
dessins}. Their underlying graph illustrates the incidence structure of finite
projective spaces $\PR{m}{n}$. Usually, the automorphism group of these dessins
is a cyclic \textit{Singer group} $\Sigma_\ell$ permuting transitively the
vertices. However, in some cases, a second group of automorphisms $\Phi_f$
exists. It is a cyclic group generated by the \textit{Frobenius automorphism}.
We show under what conditions $\Phi_f$ is a group of automorphisms acting
freely on the edges of the considered dessins.
