Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the *plain* sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are homotopic to the identity $f\circ g \simeq Id\_Y$, $g\circ f \simeq Id\_X$. 


Is there something of an obstruction theory that tells you when you can promote $f$ to a homotopy-equivalence in the category of $G$-spaces? 


Ideally the level of generality I care for is $X$ and $Y$ manifolds and $G$ a compact Lie group. But anything in that ballpark interests me. 



An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences?
When you say $f$ is a map of $G$-spaces I am guessing you mean a morphism in the category of $G$-spaces, i.e. a continuous map satisfying $f(ax)=af(x)$ for $a\in G$. If so, then there is a good answer. (But "promoting" the map to a strong $G$-equivalence is a funny way to say it, because it suggests a piece of extra structure rather than a property.) 


The fixed point spaces of closed subgroups are the key to much of equivariant homotopy theory. In the most important model structure for $G$-spaces, a morphism $X\to Y$ is called a weak equivalence (resp. fibration) iff for every subgroup $H$ the induced map $X^H\to Y^H$ of fixed point spaces is a weak homotopy equivalence (resp. Serre fibration). Cofibrations are generated by attaching orbits of cells. For cofibrant objects (and this includes manifolds with smooth action) weak equivalences are then the same as maps that actually have an inverse up to $G$-homotopy.


