We show that if
G
F
H
GFH
is a double Frobenius group with “upper” complement
H
H
of order
q
q
such that
C
G
(
H
)
C_G(H)
is nilpotent of class
c
c
, then
G
G
is nilpotent of
(
c
,
q
)
(c,q)
-bounded class. This solves a problem posed by Mazurov in the Kourovka Notebook. The proof is based on an analogous result on Lie rings: if a finite Frobenius group
F
H
FH
with kernel
F
F
of prime order and complement
H
H
of order
q
q
acts on a Lie ring
K
K
in such a way that
C
K
(
F
)
=
0
C_K(F)=0
and
C
K
(
H
)
C_K(H)
is nilpotent of class
c
c
, then
K
K
is nilpotent of
(
c
,
q
)
(c,q)
-bounded class.