Let \(\hat K_{2n}\) denote the complete graph \(K_{2n}\) with the edges in a perfect matching removed and let \(\hat K_{2n+1} = K_{2n+1}\). A Hamiltonian cycle system (HCS) of order \(v\) is a decomposition of the edge set of \(\hat K_v\) into a disjoint union of Hamiltonian cycles. Such a system \(H\) is called 1-rotational (resp. 2-pyramidal) if there exists \(G \leq\mathrm{Aut}(H)\) of order \(v-1\) (resp. \(v-2\)) fixing one vertex (resp. two vertices). The authors show that there is a natural construction which transforms a 1-rotational HCS of order \(2n+1\) into a 2-pyramidal HCS of order \(2n+2\) for the same group \(G\), namely, inserting a vertex in each cycle as far away as possible from the fixed vertex. They deduce that if \(n>2\) and \(H\) is a 2-pyramidal HCS of order \(2n+2\) under \(G\), then \(G =\mathrm{Aut}(H)\). They also show that the above transformation induces a two-to-one surjective mapping from the set of isomorphism classes of 1-rotational HCSs of order \(2n+1\) to the set of isomorphism classes of 2-pyramidal HCSs of order \(2n+2\). As a corollary, they obtain a lower bound for the number of isomorphism classes of 2-pyramidal HCSs.