Induced class function from conjugacy-closed normal subgroup is index of subgroup times class function inside the subgroup and zero outside the subgroup
From Groupprops
Statement
Suppose is a finite group and
is a Conjugacy-closed normal subgroup (?) (i.e.,
is both a conjugacy-closed subgroup and normal subgroup) of
. Suppose
is a Class function (?) on
with values in a field
. Then, the Induced class function (?)
is as follows:
Here, is the index of
in
.
In particular, any Direct factor (?) or any Central factor (?) is a conjugacy-closed normal subgroup, so the above applies.