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In this paper we find the structure for the automorphism group of a split metacyclic 2-group G. It can be seen as a continuation of the paper (Curran in Arch. Math. 89 (2007), 10-23) and it makes it complete. We propose a different approach to the problem than in the paper (Curran in Arch. Math. 89 (2007), 10-23). Our intention is to show that apart from some cases of 2-groups AutG has a structure similar to that of a direct product of two groups with no common direct factor [which was considered in

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we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or H-subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PSTgroups.