Let $G$ be a Hausdorff group and $X, Y$ two $G$-spaces. Assume that $f,g:X\to Y$ are two $G$-maps which are $G$-homotopic. If we consider the fixed point functor $(\cdot)^H$ for a (closed) subgroup $H$ of $G$ then it is clear that $f^H, g^H$ are homotopic.
Does the converse hold? I.e. assume that $f,g:X\to Y$ are two $G$-maps such that $f^H, g^H$ are homotopic for all closed subgroups $H$ of $G$. Does it follow that $f,g$ are $G$-homotopic? Can we construct an equivariant homotopy from non-equivariant homotopies of fixed points? If not then what if we additionally assume that $G$ is compact or even finite?