  A smooth complex projective curve is called pseudoreal if it is isomorphic to
its conjugate but is not definable over the reals. Such curves, together with
real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$.
This paper deals with the classification of pseudoreal curves according to the
structure of their automorphism group. We follow two different approaches
existing in the literature: one coming from number theory, dealing more
generally with fields of moduli of projective curves, and the other from
complex geometry, through the theory of NEC groups. Using the first approach,
we prove that the automorphism group of a pseudoreal Riemann surface $X$ is
abelian if $X/Z({\rm Aut}(X))$ has genus zero, where $Z({\rm Aut}(X))$ is the
center of ${\rm Aut}(X)$. This includes the case of $p$-gonal Riemann surfaces,
already known by results of Huggins and Kontogeorgis. By means of the second
approach and of elementary properties of group extensions, we show that $X$ is
not pseudoreal if the center of $G={\rm Aut}(X)$ is trivial and either ${\rm
Out}(G)$ contains no involutions or ${\rm Inn}(G)$ has a group complement in
${\rm Aut}(G)$. This extends and gives an elementary proof (over $\mathbb C$)
of a result by D\`ebes and Emsalem. Finally, we provide an algorithm,
implemented in MAGMA, which classifies the automorphism groups of pseudoreal
Riemann surfaces of genus $g\geq 2$, once a list of all groups acting for such
genus, with their signature and generating vectors, are given. This program,
together with the database provided by J. Paulhus in \cite{Pau15}, allowed us
to classifiy pseudoreal Riemann surfaces up to genus $10$, extending previous
results by Bujalance, Conder and Costa.
