rt.representation theory - 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.)