Let $G$ be a topological (or simplicial) group, let $X$ and $Y$ be $G$-spaces, and let $f,f':X\to Y$ be $G$-maps which are homotopic as maps of spaces. In general, $f$ and $f'$ may (of course) fail to be equivariantly homotopic. For instance, we can just take $X$ to be a point and $Y$ to be any path-connected $G$-space having two distinct fixed points of the $G$-action, with $f$ and $f'$ given by the inclusion of two fixed points. But **is there any example of such $f$ and $f'$ if we assume that $X$ and $Y$ are total spaces of principal $G$-bundles?**



Homotopic but not equivariantly homotopic maps
For any $G$-space the $G$-equivariant maps $[EG,X]\_G$, also known as the homotopy fixed points $X^{hG}$, are a Borel homotopy invariant of $X$ meaning that it is an invariant of $G$-equivariant maps for which the underlying map is a weak equivalence. If $Z$ has the trivial action, this means we can compute $[EG,Z \times EG]\_G$ as $[EG,Z]\_G=[EG/G,Z]=[BG,Z].$ On the other hand, the nonequivariant maps $[EG,Z] \cong \pi\_0(Z)$ because $EG$ is contractible. So it suffices to find a space, $Z$ such that $[BG,Z] \not= \pi\_0(Z)$. The universal example (if $G$ is not contractible) is $Z=BG$.


Explicitly, if $q:EG \rightarrow BG$ is the quotient, this produces homotopic maps $EG \xrightarrow{q \times \mathrm{Id}} BG \times EG$ and $EG \xrightarrow{\* \times \mathrm{Id}} BG \times EG$ which are not equivariantly homotopic.


