Our main goal is to determine the general solution of the functional equation
\[
f
1
(
x
y
)
+
f
2
(
x
y
−
1
)
=
f
3
(
x
)
+
f
4
(
y
)
+
f
5
(
x
)
f
6
(
y
)
,
f
i
(
t
x
y
)
=
f
i
(
t
y
x
)
(
i
=
1
,
2
)
\begin {array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_3}(x) + {f_4}(y) + {f_5}(x){f_6}(y),} \\ {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \\ \end {array}
\]
where
f
i
{f_i}
are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao’s quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d’Alembert and Wilson equations.