The paper extends the analysis of the entropies of the Poisson distribution with parameter λ. It demonstrates that the Tsallis and Sharma–Mittal entropies exhibit monotonic behavior with respect to λ, whereas two generalized forms of the Rényi entropy may exhibit “anomalous” (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as λ→∞ and provide both lower and upper bounds for them.