Abstract
A group G is called Galois-theoretical if CGCA(H)=H for any subgroup H of G and CACG(B)=B for any subgroup B of A=Aut(G). This paper shows that a group G is Galois-theoretical if and only if G is isomorphic to the trivial group, to the cyclic group of order 3, or to the symmetric group of degree 3.