Which inclusions of finite groups are relatively linearly primitive?

Question

This post is a sequel of: Which finite groups have faithful complex irreducible representations?

A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation.

An inclusion of finite groups $(H \subset G)$ is called relatively linearly primitive if there is an irreducible complex representation $V$ of $G$ such that $ker(\pi_V) \subset H$.

Question: Which inclusions of finite groups are relatively linearly primitive?

Answer 1

This answer was inspired by a comment of Dima Pasechnik.

Proposition: An inclusion of finite groups $(H \subset G)$ is relatively linearly primitive iff $G /H_G$ is linearly primitive, with $H_G$ the normal core of $H$ in $G$.

Proof: If $(H \subset G)$ is relatively linearly primitive, then there is an irreducible complex representation $V$ of $G$ such that $ker(\pi_V) \subset H$, but $ker(\pi_V)$ is normal in $G$ so by definition $ker(\pi_V) \subset H_G$, so that $W=V^{H_G}$ is an irreducible faithful representation of $G/H_G$.
Conversly, if $G /H_G$ is linearly primitive then it has a faithful complex irreducible representation $W$, on which $G$ acts irreducibly with $ker(\pi_W) = H_G$, but $H_G \subset H$, so the result follows. $\square$