Let
G
⊂
G
L
(
C
r
)
G\subset \mathrm {GL}(\mathbb {C}^r)
be a finite complex reflection group. We show that when
G
G
is irreducible, apart from the exception
G
=
S
6
G=\mathfrak {S}_6
, as well as for a large class of non-irreducible groups, any automorphism of
G
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
G
L
(
C
r
)
(
G
)
N_{\mathrm {GL}(\mathbb {C}^r)}(G)
and of a “Galois” automorphism: we show that
G
a
l
(
K
/
Q
)
\mathrm {Gal}(K/\mathbb {Q})
, where
K
K
is the field of definition of
G
G
, injects into the group of outer automorphisms of
G
G
, and that this injection can be chosen such that it induces the usual Galois action on characters of
G
G
, apart from a few exceptional characters; further, replacing
K
K
if needed by an extension of degree
2
2
, the injection can be lifted to
A
u
t
(
G
)
\mathrm {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
G
can be chosen rational.