Let $\rho$ be an irreducible representation of a group $N$, and let $G,H$ be groups with $N$ of finite index in $H$ and $H$ normal in $G$. Let $\pi=\rho^H$ be the induced representation of $\rho$ to $H$; I'd like to understand the isotropy of $\pi$ in $G$, that is, 
$I\_G(\pi)=\lbrace g\in G:\pi^g\sim \pi\rbrace$, those $g$ such that $\pi$ and $h\mapsto\pi(g^{-1}hg)$ are equivalent.


Guess: if $I\_H(\rho)=N$, that is, if $\pi$ is irreducible, then
$$I\_G(\pi)=H\cdot I\_G(\rho).$$


It's easy to see that "$\supseteq$" holds.



Isotropy (aka inertia) of induced representation
The "guess" is wrong; here is a counterexample. Take $G$ to be dihedral of order 16. Let $H$ be one of the two copies of the dihedral group of order 8 in $G$, and let $N$ be one of the two copies of the Klein fours group in $H$. Let $\pi$ be the unique irreducible character of degree $2$ of $H$, and let $\rho$ be one of the two linear constituents of the restriction $\pi\_N$.


Now $\rho^H = \pi$ is irreducible, and since $\pi$ is unique, it is invariant in $G$. We argue, however, that $G \ne HT$, where $T = I\_G(\rho)$. By definition, $T$ is contained in the normalizer in $G$ of $N$, and of course, $N$ is also normal in $H$. Thus if $G = HT$, it would follow that $N$ is normal in $G$. This is not the case, however. One way to see that
$N \not\kern -2pt\triangleleft\ G$ is to observe that otherwise $G/C$ would be embedded in
${\rm Aut}(G)$, where $C$ is the centralizer of $N$. This would force $|C| \ge 8$, and that would imply that $N$ is contained in an abelian subgroup of $G$ of order $8$. The only abelian subgroup of oirder $8$ in $G$, however, is cyclic, so does not contain $N$.


