If $H$ has a normal complement then it has the character restriction property

Question

I am trying to show that if a subgroup $H$ of a group $G$ has a normal complement then $H$ has the character restriction property. That is every irreducible character of $H$ is a restriction of some irreducible character of $G.$

I started with an irreducible character $\theta$ of $H$ and tried to construct $\chi \in Irr(G)$ such that $\chi_H=\theta$ using the definition of induced character. I also tried using the fact that if $HK=G$ and $\varphi \in \text{cf}(H)$ then $$(\varphi^{G})_K=(\varphi_{H \cap K})^K.$$ But I could not complete it.

Answer 1

I have found the answer. Here is my proof:

If $\theta$ is an irreducible character of $H$ then define $\chi(g)=\chi(hk)=\theta(h).$ Then

\begin{align} [\chi, \chi] &= \dfrac{1}{|G|} \sum\limits_{g \in G} \chi(g) \overline{\chi(g)} \\ &= \dfrac{1}{|G|} \sum\limits_{hk \in HK} \chi(hk) \overline{\chi(hk)} \\ &= \dfrac{|K|}{|G|} \sum\limits_{h \in H} \theta(h) \overline{\theta(h)}\\ &= \dfrac{1}{|H|} \sum\limits_{h \in H} \theta(h) \overline{\theta(h)} \\ &=[\theta, \theta] \end{align}

Hence $\chi$ is an irreducible character of $G$ with $\chi_H=\theta.$