If $X$ is a vertex-transitive graph and the stabilizer of a vertex has order $m$, then the lexicographic product of $K_m$ by $X$ is a Cayley graph. We get the lexicographic product here by replacing each vertex of $X$ by $K_m$ and, where two vertices of $X$ are adjacent, join each vertex in one $K_m$ to each vertex in the other. So this product contains copies of $X$ as an induced subgraph and is a Cayley graph for $Aut(X)$. The result and the construction are due to Sabidussi.
If $X$ is a vertex-transitive graph and the stabilizer of a vertex has order $m$, then the lexicographic product of $K_m$ by $X$ is a Cayley graph. We get the lexicographic product here by replacing each vertex of $X$ by $K_m$ and, where two vertices of $X$ are adjacent, join each vertex in one $K_m$ to each vertex in the other. So this product contains copies of $X$ as an induced subgraph and is a Cayley graph for $Aut(X)$. The result and the construction are due to Sabidussi.