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In 2018 we proved that if $G$ is non-cyclic group of order $n$, then $\\psi(G)\\leq \\frac 7{11}\\psi(C_n)$ and equality holds if $n=4k$ with $(k,2)=1$ and $G=(C_2\\times C_2)\\times C_k$. In this paper we proved that equality holds if and only if $n$ and $G$ are as indicated above. 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