Languages invariant under more symmetries: overlapping factors versus palindromic richness

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $$\mathcal{P}$(n) + $\mathcal{P}$(n+1) \leq 2 + $\mathcal{C}$(n+1)-$\mathcal{C}$(n)$forany$n\in $\mathbb{N}$$,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.