Let
T
T
be an ergodic automorphism of a Lebesgue space and
α
\alpha
a cocycle of
T
T
with values in an Abelian locally compact group
G
G
. An automorphism
θ
\theta
from the normalizer
N
[
T
]
N[T]
of the full group
[
T
]
[T]
is said to be compatible with
α
\alpha
if there is a measurable function
φ
:
X
→
G
\varphi : X \to G
such that
α
(
θ
x
,
θ
T
θ
−
1
)
=
−
φ
(
x
)
+
α
(
x
,
T
)
+
φ
(
T
x
)
\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)
at a.e.
x
x
. The topology on the set
D
(
T
,
α
)
D(T, \alpha )
of all automorphisms compatible with
α
\alpha
is introduced in such a way that
D
(
T
,
α
)
D(T , \alpha )
becomes a Polish group. A complete system of invariants for the
α
\alpha
-outer conjugacy (i.e. the conjugacy in the quotient group
D
(
T
,
α
)
/
[
T
]
)
D(T, \alpha )/[T])
is found. Structure of the cocycles compatible with every element of
N
[
T
]
N[T]
is described.