For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x, y\in X\text{ and } x\neq y\big\}$.

Let $G=(V,E)$ be a simple, undirected graph, and suppose ${\cal V}$ is a collection of subsets of $V$ such that for all $W, W' \in {\cal V}$ we have either $W\subseteq W'$ or $W' \subseteq W$.

If for all $W\in {\cal V}$ the induced subgraph $(W,E\cap [W]^2)$ is vertex-transitive, is the induced subgraph $(\bigcup {\cal V}, E\cap [\bigcup{\cal V}]^2)$ necessarily vertex-transitive as well?

(Applying Zorn's Lemma, which is equivalent to the Axiom of Choice, a positive answer would imply that every graph has a maximal vertex-transitive subgraph.)