Let $H \leq G$ be a subgroup, where $G$ is finite. Suppose $(\sigma, W)$ is an irreducible representation of $H$. Let $\sigma^{\circ}$ be the induced representation on $G$. Maschke's theorem tells us that $\sigma^{\circ} = W_1^{a_1} \oplus W_2^{a_2} \oplus \dotsb \oplus W_n^{a_n}$, where each $W_i$ is an irreducible subrepresentation of $W$. So taking characters, we have $\chi_{\sigma^{\circ}} = a_1 \chi_{1} + \dotsb + a_n \chi_{n}$.
I am trying to show that $$ a_1^{2} + \dotsb + a_{n}^2 \leq \frac{|G|}{|H|} . $$
So $\sum_{i=1}^n a_i^{2} = \langle \chi_{\sigma^{\circ}}, \chi_{\sigma^{\circ}} \rangle = \langle \chi_{\sigma}, \operatorname{Res}^{G}_{H} \chi_{\sigma^{\circ}} \rangle$. Where the last equality holds by Frobenius reciprocity.
I am not sure how one proceeds. I suppose if $\operatorname{Res}^{G}_{H} \chi_{\sigma^{\circ}} = \chi_{\sigma}$ then the result is straightforward.
You’ve reduced the problem to showing that the multiplicity of $\chi$ in $Res^G_H(Ind^G_H(\chi))$ is less than or equal to the index $[G:H]$. For this, note that the process of inducing then restricting from $H$ to $G$ multiplies dimensions of representations by $[G:H]$, so at most $[G:H]$ “distinct copies” of $W$ can fit inside this restricted, induced representation. Precisely, if the multiplicity of the irrep $W$ in $Res^G_HInd^G_H(W)$ is $m$, then $$m\cdot dim(W)\leq dim(Res^G_HInd^G_H(W))$$ and this upper bound is $[G:H]dim(W)$, giving $m\leq [G:H]$.