Does any vertex-transitive graph $G$ have a vertex-transitive line graph $L(G)$?

Question

Is it true that any vertex-transitive graph $G$ has a vertex-transitive line graph $L(G)$? It seems to me that if I proved that for any $$e_1=\{x,y\},e_2=\{a,z\} \in E(G)$$ there exists an automorphism $\rho\in \text{Aut}(G)$ s.t. $$x=\rho(a) \wedge y=\rho(z) (1)$$ I would have a positive answer. There is probably a connection between the automorphisms' group $\text{Aut}(G)$ too (all are conjugate. Probably not a very helpful notice), but I think that there must be counterexample where $(1)$ doesn't hold. Could anyone help me clear this out?

Answer 1

From https://mathworld.wolfram.com/Vertex-TransitiveGraph.html:

'A undirected connected graph is edge-transitive if its line graph is vertex-transitive'

Thus, if for any vertex-transitive graph $G$, the graph $L(G)$ is also vertex-transitive, it means that any vertex-transitive graph is edge-transitive.

This is false, see for a counter examples and more information: Graphs that are vertex transitive but not edge transitive