groups with some central automorphisms fixing the central kernel quotient

Let g be a group. an automorphism α of the group g is called a central automorphism, if x−1x α ∈ z(g) for all x ∈ g. let lc(g) be the central kernel of g, that is, the set of elements of g fixed by all central automorphisms of g and autlc (g) denote the group of all central automorphisms of g fixing g/lc(g) element-wise. in the present paper, we investigate the properties of such automorphisms. moreover, a full classification of p-groups g of order at most p 5 where autlc (g) = inn(g) is also given.