Graphs with automorphism $D_n \times G$ where $G$ is any finite permutation group

Question

I would like to define a systematic process to construct graphs with automorphism $D_n \times G$ where $G$ is any finite permutation group. Here, $D_n$ is the dihedral group of order $2 n$.

I follow the following process suggested by Morgan Rodgers.

1. Construct cycle $C_n$. The automorphism group of this cycle is $D_n$
2. Add the same subgraph to every nodes of $C_n$.

How can I rigorously prove that the automorphism group of any graph created by this process is the direct product of a finite group and a dihedral group $D_n$?