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We study the finite frame property of some extensions of Fitting, Marek, andTruszczy\'nski's pure logic of necessitation $\mathbf{N}$. For any naturalnumbers $m, n$, we introduce the logic $\mathbf{N}^+\mathbf{A}_{m, n}$ byadding the single axiom scheme $\Box^n \varphi \to \Box^m \varphi$ and the rule$\dfrac{\neg \Box \varphi}{\neg \Box \Box \varphi}$ (Ros$^\Box$) into$\mathbf{N}$. We prove the finite frame property of $\mathbf{N}^+\mathbf{A}_{m,n}$ with respect to Fitting, Marek, and Truszczy\'nski's relational semantics.We also prove that the Ros$^\Box$ rule is necessary for the completeness of$\mathbf{N}^+\mathbf{A}_{0, n}$ for $n \ge 2$.

The finite frame property of some extensions of the pure logic of necessitation