For a normal subgroup
N
N
of the free group
F
d
\mathbb {F}_{d}
with at least two generators, we introduce the radial limit set
Λ
r
(
N
,
Φ
)
\Lambda _{r}(N,\Phi )
of
N
N
with respect to a graph directed Markov system
Φ
\Phi
associated to
F
d
\mathbb {F}_{d}
. These sets are shown to provide fractal models of radial limit sets of normal subgroups of Kleinian groups of Schottky type. Our main result states that if
Φ
\Phi
is symmetric and linear, then we have that
dim
H
(
Λ
r
(
N
,
Φ
)
)
=
dim
H
(
Λ
r
(
F
d
,
Φ
)
)
\dim _{H}(\Lambda _{r}(N,\Phi ))=\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))
if and only if the quotient group
F
d
/
N
\mathbb {F}_{d}/N
is amenable, where
dim
H
\dim _{H}
denotes the Hausdorff dimension. This extends a result of Brooks for normal subgroups of Kleinian groups to a large class of fractal sets. Moreover, we show that if
F
d
/
N
\mathbb {F}_{d}/N
is non-amenable, then
dim
H
(
Λ
r
(
N
,
Φ
)
)
>
dim
H
(
Λ
r
(
F
d
,
Φ
)
)
/
2
\dim _{H}(\Lambda _{r}(N,\Phi ))>\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))/2
, which extends results by Falk and Stratmann and by Roblin.