  Using the description of the Frobenius limit of modules over the ring of
invariants under an action of a finite group on a polynomial ring over a field
of characteristic $p>0$ developed by Symonds and the author, we give a
characterization of the ring of invariants with a positive dual $F$-signature.
Combining this result and Kemper's result on depths of the ring of invariants
under an action of a permutation group, we give an example of an $F$-rational,
but non-$F$-regular ring of invariants under the action of a finite group.
