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 $\mathrm{GL}(V)$. If $V$ is a permutation representation of $G$, or more generally a monomial representation, we prove that the ring of invariants $(\mathrm{Sym}V{)}^G$ 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 ${\mathbb{F}}_3(t)$, and $G$ an elementary abelian subgroup of $\mathrm{GL}(V)$, such that the invariant ring $(\mathrm{Sym}V{)}^G$ does not have finite Frobenius representation type.