Summary: We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible \(G\)-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a category. This construction interpolates in a sense between “framed” and “colored” versions of operads; we hope that it will be of independent interest. An algebra over the cohomology of this operad is the same thing as a \(G\)-equivariant CohFT, as defined by T. J. Jarvis et al. [Compos. Math. 141, No. 4, 926–978 (2005; Zbl 1091.14014)]. We prove that the (orbifold) Gromov-Witten invariants of global quotients \([X/G]\) give examples of \(G\)-CohFTs.