Let $G$ be a group and $\mathcal{N}$ be the class of all nilpotent groups. A subset $A$ of $G$ is said to be nonnilpotent if for any two distinct elements $a$ and $b$ in $A$, $\langle a, b\rangle \not\in \mathcal{N}$. If, for any other nonnilpotent subset $B$ in $G$, $|A|\geq |B|$, then $A$ is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by $\omega(\mathcal{N}_G)$. In this paper, among other results, we obtain $\omega(\mathcal{N}_{Suz(q)})$ and $\omega(\mathcal{N}_{PGL(2,q)})$, where $Suz(q)$ is the Suzuki simple group over the field with $q$ elements and $PGL(2,q)$ is the projective general linear group of degree $2$ over the finite field with $q$ elements, respectively.

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