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 $X=G/H$.

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 nontrivial $H$fixed quotient. (If $H$ is finite and $F$ has characteristic zero, or at least prime to the order of $H$, so that $\rho$ is semisimple as an $H$representation, then this is equivalent to requiring that $\rho$ have a nontrivial $H$fixed subrepresentation.) (Note also that a nonzero $G$equivariant map out of $\rho$ is automatically injective, because $\rho$ is irreducible.) 

