Let $H_{X}\subset H_{Y} \subset H_{K}$ be closed subgroups of a topological group $G$. Let $X = G/H_{X}$ $Y = G/H_{Y}$, $K = G/H_{K}$. Now, suppose we have the following diagram: $$\require{AMScd}\begin{CD} X @>{\phi}>> Y\\ @ViVV @VVjV\\ K @>{Id}>> K \end{CD}$$ Where $i$ and $j$ are both locally trivial fiber bundles and $\phi$ is equivarient w.r.t. the action of $G$. Does $\phi$ have the homotopy lifting property?
I have shown that given a homotopy $H:W\times I \rightarrow Y$ that there exist a map $\tilde{H}:W\times I \rightarrow X$ such that $H$ is homotopic to $\phi\circ \tilde{H}$ relative to $W\times {0}$. I wanted to know if this is good enough for the LES of homotopy groups?