Let [Formula: see text] be a finite [Formula: see text]-group. In our recent paper, it was shown that in a finite [Formula: see text]-group of almost maximal class, the set of all commuting automorphisms, [Formula: see text] is a subgroup of [Formula: see text]. Also, we proved that the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. Using these results, in this paper, we will take of the task of determining when the group of all commuting automorphisms of all finite [Formula: see text]-groups of almost maximal class are equal to the group of all central automorphisms. This determination is not easy. We will prove they are equal, except only for five ones. We show that the minimum order of a [Formula: see text]-group which it’s group of all commuting automorphisms is not equal to it’s group of all central automorphisms is [Formula: see text]. Also, we prove that if [Formula: see text] is a finite [Formula: see text]-group in which [Formula: see text], then the subgroup of right 2-Engel elements of [Formula: see text], [Formula: see text], coincides with the second term of upper central series of [Formula: see text].

Let Γ(G) denote the set of commutators of a group G, G′ = ⟨Γ(G)⟩ and c(G) (or cw(G)) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. Recently we proved that if G is a finite p-group of maximal cl

Let [Formula: see text] be a group. If the set [Formula: see text] for all [Formula: see text] forms a subgroup of [Formula: see text], then [Formula: see text] is called [Formula: see text]-group. Let [Formula: see text] be an odd prime. Recently it has been proven that a finite [Formula: see text]-group of almost maximal class is an [Formula: see text]-group. For finite 2-groups of almost maximal class, the situation is much more complicated. This paper deals with the case [Formula: see text]. We prove that a 2-group of almost maximal class is an [Formula: see text]-group. We show the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. We also discuss the direct product of certain [Formula: see text]-groups and subsequently we give some applications of our results.

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