Frobenius kernel criterion by irreducible induction (${\mathrm{Ind}}_N^G\phi \in \mathrm{Irr}(G)(1_N\ne \phi \in \mathrm{Irr}(N))$)

For $1<N◃G$ proper, the displayed condition is equivalent to $N$ being a Frobenius kernel. For a normal subgroup the induction norm is $|I_G(\phi ):N|$. Trivial inertia for each nonprincipal character and Brauer's permutation lemma imply that every element outside $N$ has trivial centralizer in $N$. This forces $N$ to be a normal Hall subgroup; a complement is then fixed-point-free on $N$.