Induced class function from conjugacy-closed normal subgroup is index of subgroup times class function inside the subgroup and zero outside the subgroup

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Statement

Suppose G is a finite group and H is a Conjugacy-closed normal subgroup (?) (i.e., H is both a conjugacy-closed subgroup and normal subgroup) of G. Suppose \theta is a Class function (?) on H with values in a field k. Then, the Induced class function (?) \operatorname{Ind}_H^G \theta is as follows:

\operatorname{Ind}_H^G \theta(x) = \lbrace\begin{array}{rl}[G:H]x, & x \in H \\ 0, & x \notin H \\\end{array}

Here, [G:H] is the index of H in G.

In particular, any Direct factor (?) or any Central factor (?) is a conjugacy-closed normal subgroup, so the above applies.

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