Let $\Gamma$ be a vertex transitive graph with $4p$ vertices, where $p$ is an odd prime. Let $\Delta$ be an imprimitive block of lenght $p$ for the automorphism group of the graph. How can I prove that $\Gamma$ is isomorphic to the lexicographic product of a graph with 4 vertices and a graph isomorphic to the subgraph of $\Gamma$ on $\Delta$? I know that by one of the C. Godsil's papers (On Cayley graph isomorphisms, Ars Comb. 1983) when $\Gamma$ is a Cayley graph over $\Bbb Z_{4p}$ and $\Delta$ is the support of a $p$-cycle, the result is true. Godsil claimed in hers paper that this is an easy exercise. But unfortunately I can not prove it. I guess that the proof of the Godsil's claim can be useful to prove my calim.

Let $\Gamma$ be a vertex transitive graph with $4p$ vertices, where $p$ is an odd prime. Let $\Delta$ be an imprimitive block of length $p$ for the automorphism group of the graph. How can I prove that $\Gamma$ is isomorphic to the lexicographic product of a graph with 4 vertices and a graph isomorphic to the subgraph of $\Gamma$ on $\Delta$? I know that by one of C. Godsil's papers (On Cayley graph isomorphisms, Ars Comb. 1983) when $\Gamma$ is a Cayley graph over $\Bbb Z_{4p}$ and $\Delta$ is the support of a $p$-cycle, the result is true. Godsil claimed in her paper that this is an easy exercise. But unfortunately I cannot prove it. I guess that the proof of Godsil's claim can be useful to prove my claim.