  Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that
when $G$ is irreducible, apart from the exception $G=\Sgot_6$, as well as for a
large class of non-irreducible groups, any automorphism of $G$ is the product
of a central automorphism and of an automorphism which preserves the
reflections. We show further that an automorphism which preserves the
reflections is the product of an element of $N_{\GL(\BC^r)}(G)$ and of a
"Galois" automorphism: we show that $\Gal(K/\BQ)$, where $K$ is the field of
definition of $G$, injects into the group of outer automorphisms of $G$, and
that this injection can be chosen such that it induces the usual Galois action
on characters of $G$, apart from a few exceptional characters; further,
replacing if needed $K$ by an extension of degree 2, the injection can be
lifted to $\Aut(G)$, and every irreducible representation admits a model which
is equivariant with respect to this lifting. Along the way we show that the
fundamental invariants of $G$ can be chosen rational.
