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
