Factor frequencies in languages invariant under more symmetries

The number of frequencies of factors of length $n+1$ in a recurrent aperiodic infinite word does not exceed $3\mathrm{\Delta }\text{C}(n)$, where $\mathrm{\Delta }\text{C}(n)$ is the first difference of factor complexity, as shown by Boshernitzan. Pelantov'a together with the author derived a better upper bound for infinite words whose language is closed under reversal. In this paper, we further diminish the upper bound for uniformly recurrent infinite words whose language is invariant under all elements of a finite group of symmetries and we prove the optimality of the obtained upper bound.