A finite group G has a self-centralization system of type
(
2
|
A
1
|
,
4
|
A
2
|
,
4
|
A
3
|
)
(2|{A_1}|,4|{A_2}|,4|{A_3}|)
if G contains three nonconjugate CC-subgroups
A
1
,
A
2
,
A
3
{A_1},{A_2},{A_3}
, such that
|
N
G
(
A
1
)
|
=
2
|
A
1
|
,
|
N
G
(
A
2
)
|
=
4
|
A
2
|
,
|
N
G
(
A
3
)
|
=
4
|
A
3
|
|{N_G}({A_1})| = 2|{A_1}|,|{N_G}({A_2})| = 4|{A_2}|,|{N_G}({A_3})| = 4|{A_3}|
. The authors prove that if a finite group G has a self-centralization system of type
(
2
|
A
1
|
,
4
|
A
2
|
,
4
|
A
3
|
)
(2|{A_1}|,4|{A_2}|,4|{A_3}|)
and
|
G
|
⩽
3
|
A
1
|
2
|
A
2
|
2
|
A
3
|
2
|G| \leqslant 3|{A_1}{|^2}|{A_2}{|^2}|{A_3}{|^2}
, then G has a nilpotent normal subgroup N such that G/N is isomorphic to
S
z
(
q
)
Sz(q)
for suitable q.