Palindromic richness for languages invariant under more symmetries

For a given finite group G   consisting of morphisms and antimorphisms of a free monoid A⁎A⁎, we study infinite words with language closed under the group G. We focus on the notion of G-richness which describes words rich in generalized palindromic factors, i.e., in factors w   satisfying Θ(w)=wΘ(w)=w for some antimorphism Θ∈GΘ∈G. We give several equivalent descriptions which are generalizations of known characterizations of rich words (in the terms of classical palindromes) and show two examples of G-rich words.