Let (R, S) be a distributively generated near ring satisfying
(
R
,
S
)
⊆
(
E
(
G
)
,
End
(
G
)
)
(R,S) \subseteq (E(G),{\text {End}}(G))
and
S
⊆
End
(
G
)
S \subseteq {\text {End}}(G)
for some group G, endomorphism near ring
E
(
G
)
E(G)
, and subsemigroup S of the endomorphisms of G,
End
(
G
)
{\text {End}}(G)
. The radicals
J
(
R
)
J(R)
of (R, S) are characterized in terms of series of subgroups of G. We assume S contains the inner automorphisms of G and obtain two main results on characterizing series. (1) If G satisfies both chain conditions on S subgroups then a unique minimal characterizing series exists. (2) If G is finite, then both maximal and minimal characterizing series exist, are unique, and are themselves characterized in G.