Frobenius representation type for invariant rings of finite groups

Let V be a finite rank vector space over a perfect field of characteristic $p>0$, and let G be a finite subgroup of $\mathrm{GL}\left(V\right)$. If V is a permutation representation of G, or more generally a monomial representation, we prove that the ring of invariants ${\left(\mathrm{Sym}V\right)}^G$ has finite Frobenius representation type. We also construct an example with V a finite rank vector space over the algebraic closure of the function field $F_3\left(t\right)$, and G an elementary abelian subgroup of $\mathrm{GL}\left(V\right)$, such that the invariant ring ${\left(\mathrm{Sym}V\right)}^G$ does not have finite Frobenius representation type.