We define a model category structure on a slice category of simplicial spaces, called the "Segal group action" structure, whose fibrant-cofibrant objects may be viewed as representing spaces $X$ with an action of a fixed Segal group (i.e. a group-like, reduced Segal space). We show that this model structure is Quillen equivalent to the projective model structure on $G$-spaces, $S^BG}$, where $G$ is a simplicial group corresponding to the Segal group. One advantage of this model is that if we start with an ordinary group action $X\in S^BG$ and apply a weakly monoidal functor of spaces $L: S \to S$ (such as localization or completion) on each simplicial degree of its associated Segal group action, we get a new Segal group action of $LG$ on $LX$ which can then be rigidified via the above-mentioned Quillen equivalence.
Keywords: Model category, Segal space, group action, equivariant homotopy theory
2010 MSC: 55U35
Theory and Applications of Categories, Vol. 30, 2015, No. 40, pp 1287-1305.