Let $\rho$ be irreducible representation of group $G$. 
How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$. 



Embedding into Permutation Representation
There is the following adjointness (a form of Frobenius reciprocity):


$Hom\_G(\rho,F^X) = Hom\_H(\rho,trivial).$


Thus $\rho$ embeds in $F^X$ if and only if $\rho$ admits a non-trivial $H$-fixed quotient.


(If $H$ is finite and $F$ has characteristic zero, or at least prime to the order
of $H$, so that $\rho$ is semi-simple
as an $H$-representation, then this is equivalent to requiring that $\rho$ have a 
non-trivial $H$-fixed subrepresentation.)


(Note also that a non-zero $G$-equivariant map out of $\rho$ is automatically injective,
because $\rho$ is irreducible.)


