On the probability distribution associated to commutator word map in finite groups \rom{2}

Let $P(G)$ denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that $|P(G)|=1$, for any finite group $G$ of nilpotency class $3$ with exactlly two conjugacy class sizes. We also show that for given $n\geq 1$, there exists a finite group $G$ of nilpotency class $2$ with exactlly two conjugacy class sizes such that $|P(G)|=n$.