If $bF_q$ is the finite field of order $q$ and
characteristic $p$, let $F(q)$ be the category whose
objects are functors from finite dimensional $F_q$--vector spaces to
$F_q$--vector spaces,
and with morphisms the
natural transformations between such functors. Important families of objects in $F(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in F_q[\Sigma_n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma_n}$,
$ S^n(V) = V^{\otimes n}/\Sigma_n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$.
Fixing $F$, we discuss the problem of computing $Hom_{F(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $Hom_{F(q)}(S_m, G)$ for all $m$. When $q = p$, we get a complete answer for any functor $F$ chosen from the families listed above.
Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when $F=S^n$, arose in recent work on the homology of iterated loopspaces.