For a finite group $G$ with normal subgroup $H$, the induced representation $\text{Ind}_H^G(1)$ decomposes as a sum of irreducibles with the multiplicities equal to the dimensions, because it is is the pullback of the right regular representation of $G/H$. For a subgroup which is not normal, this need not be true. For example, take $G=S_3$ and $H$ a subgroup of order $2$. The index $3$ can only be written as a sum of squares as $1^2+1^2+1^2$, but $S_3$ only has two distinct $1$ dimensional representations.

Is the converse true? That is, if $\text{Ind}_H^G(1)$ decomposes into irreducible representations with multiplicity equal to the dimension, is $H$ necessarily normal in $G$? If not, is there some other characterization of subgroups which have this property?

The question arises in generalizing the 'Uncertainty Principle' for finite abelian groups $$ |G|\le \sharp \text{supp}(f)\cdot \sharp\text{supp}(\hat f), $$ to finite groups in general, and when the corresponding inequality is tight.