A torsion-free abelian group
G
G
is said to be a Butler group if
Bext
(
G
,
T
)
\operatorname {Bext} (G,\,T)
for all torsion groups
T
T
. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group
G
G
satisfies the T.E.P. over a pure subgroup
H
H
if and only if
H
H
is decent in
G
G
in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called
B
2
{B_2}
-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a
B
2
{B_2}
-group. We show under
(
V
=
L
)
(V = L)
that this is indeed the case for Butler groups of rank
ℵ
1
{\aleph _1}
. On the other hand it is shown that, under ZFC, it is undecidable whether a group
B
B
for which
Bext
(
B
,
T
)
=
0
\operatorname {Bext} (B,\,T) = 0
for all countable torsion groups
T
T
is indeed a
B
2
{B_2}
-group.