In 1982, Durnberger proved that every connected Cayley graph of a finite group with a commutator subgroup of prime order contains a hamiltonian cycle. In this paper, we extend this result to the infinite case. Additionally, we generalize this result to a broader class of infinite graphs $X$, where the automorphism group of $X$ contains a transitive subgroup $G$ with a cyclic commutator subgroup of prime order.