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We investigate the structure of finite groups of order 𝑝³𝑞, where 𝑝 and 𝑞 are distinct prime numbers. Under the arithmetic conditions 𝑞 ≠ 1 (mod 𝑝) and 𝑝 ≠1 (mod 𝑞), and assuming that the center of the group has order divisible by 𝑝², we prove that such groups must be abelian. As a consequence, we classify all possible isomorphism types using the Fundamental Theorem of Finite Abelian Groups. We also discuss the situation in which the centrality assumption is removed and explain how non-abelian examples may arise.