Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.
Is there a description of this spectrum as some sort of Thom spectrum?
I did find a construction in Costenoble's "An introduction to equivariant cobordism", but I do not understand in which sense his spectrum represents geometric $G$-cobordism. I have the feeling that it could be just the restriction of $MO_G$ to the trivial universe, but I'm getting really confused about this.