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 $Y$).