Unique Structure on G-equivariant Manifolds

We will have a deep look at the set of all $G$-equivariant maps from the factor Lie group $G$ to the under the action manifold $M$, both from “computational” and “observability” viewpoint. We will also be looking for the existence of “unique” structure on this set, in a way that the induced-action of Lie group $G$ be smooth on the mentioned set.