An extension of a normal subgroup representation

Let $G$ be finite, $N⊴G$, $N\le H\le G$, and $\rho :N\to \mathrm{GL}(S)$ an irreducible complex representation. An extension of $S$ to $H$ is a representation $\tilde{\rho }:H\to \mathrm{GL}(S)$ on the same space with $\tilde{\rho }{\mathrm{\mid }}_N=\rho$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree, The sign representation of $S_n$ and the restriction ${\mathrm{Res}}_H^G(V)$ of a representation to a subgroup). At character level an extension of $\theta ={\chi }_S$ is a character $\tilde{\theta }$ with ${\mathrm{Res}}_N^H\tilde{\theta }=\theta$.

Every extension is irreducible: any $H$-stable subspace is $N$-stable, so is zero or all of $S$. Its existence implies invariance under $H$, since $\tilde{\rho }(h)$ intertwines the conjugate action with the original one. Extension existence is an additional hypothesis in the correspondence below.

For a representation $M$ of $H\mathrm{/}N$, its inflation to $H$ is the composite with $H\to H\mathrm{/}N$. Conversely an $H$-representation on which $N$ acts trivially descends uniquely to $H\mathrm{/}N$, and irreducibility is preserved in both directions, by A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation. Inflation and extension are different constructions: an extension retains the given, possibly nontrivial, $N$-action.