  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 $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 +
\mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. 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.
