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The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group G. To achieve this, we introduce the moduli functor M h G of G -marked Gorenstein canonical models with Hilbert polynomial h , and prove the existence of 𝔐 h [ G ] , the coarse moduli scheme for M h G . Then we show that 𝔐 h [ G ] has a proper and finite morphism onto 𝔐 h so that its image 𝔐 h (G) is a closed subscheme. In the end we obtain the canonical representation type decomposition 𝒟 h [ G ] of 𝔐 h [ G ] and use 𝒟 h [ G ] to study the structure of 𝔐 h [ G ]. [ABSTRACT FROM AUTHOR]