We give a general construction of coded systems with an automorphism group isomorphic to
Z
⊕
G
\mathbf {Z}\oplus G
where
G
G
is any preassigned group which has a “continuous block presentation” (the isomorphism will map the shift to
(
1
,
e
G
)
)
(1,e_G))
. Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains
Z
[
1
/
2
]
\mathbf {Z}[1/2]
. We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.