The second maximal groups with respect to the sum of element orders

Denote by G a finite group and let ψ(G) denote the sum of element orders in G. In 2009, H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs proved that if |G|=n and G is non-cyclic, then ψ(G)<ψ(C_n), where C_n denotes the cyclic group of order n. In 2018 we proved that if G is non-cyclic group of order n, then [Formula presented] and equality holds if n=4k with (k,2)=1 and G=(C₂×C₂)×C_k. In this paper we proved that equality holds if and only if n and G are as indicated above. Moreover we proved the following generalization of this result: Theorem 4. Let G be a non-cyclic group of order n, with q being the least prime divisor of n. Then [Formula presented], with equality if and only if n=q²k with (k,q)=1 and G=(C_q×C_q)×C_k. Notice that if q=2, then [Formula presented].