  Let $V$ be a finite rank vector space over a perfect field of characteristic
$p>0$, and let $G$ be a finite subgroup of $\operatorname{GL}(V)$. If $V$ is a
permutation representation of $G$, or more generally a monomial representation,
we prove that the ring of invariants $(\operatorname{Sym}V)^G$ 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
$\operatorname{GL}(V)$, such that the invariant ring $(\operatorname{Sym}V)^G$
does not have finite Frobenius representation type.
