Let G be a group and let M and N be two normal subgroups of G. Let [Formula: see text] be the set of all automorphisms of G which centralize G/M and N. In this paper, we find certain necessary and sufficient conditions on G such that [Formula: see text] be equal to Z( Inn (G)), Inn (G), C* or Aut c(G). We also characterize the subgroups of a finite p-group G for which the equality [Formula: see text] holds.

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Let G be a group and let Aut c(G) be the group of all central automorphisms of G. Let C* = C Aut c(G)(Z(G)) be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper, we prove that if G is a finitely generated nilpotent group of class 2, then C* ≃ Inn (G) if and only if Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). Also we prove that if G is a finitely generated group which is not torsion-free, then C* = Inn (G) if and only if G is nilpotent group of class 2 and Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). In both cases, we show G has a particularly simple form.

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