Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume that $\pi_n(X/G)\cong \pi_n(Y/G)$ for all $n\geq 0$ and are induced by the cellular inclusion $Y/G\hookrightarrow X/G$.

Whitehead's Theorem implies that there is a strong deformation retraction (SDR) from $X/G$ to $Y/G$.

In this setting, does there exist a **$G$-equivariant SDR** from $X$ to $Y$?

If not, what if one *further* assumes the existence of a SDR from $X$ to $Y$ (not assumed $G$-equivariant). Would that then imply the existence of a **$G$-equivariant SDR** from $X$ to $Y$?

EDIT: After Tom Goodwillie answered both questions negatively, I have decided to add another assumption; namely, assume that the fixed point set $X^G$ is contained in $Y$ (or perhaps assume that $X^G$ $G$-equivariantly retracts to a subspace of $Y$).

`$G/H \times D^n$'. So when $G$ is compact Lie, $n$-skeleta do not have geometric dimension $n$. Subcomplex must be taken in this equivariant sense, a union of equivariant cells. Then if`

$X^H \to Y^H$` is a weak homotopy equivalence, $X\to Y$ is the inclusion of a SDR. All bets are off for other guesses as to what a $G$-CW complex means, and for orbits replacing fixed points, and for merely nonequivariant SDR's. Just not in the cards. – Peter May Jul 18 '12 at 2:18