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For a group G, a subgroup H oi G is said to be conjugate permutable if HHX = HXH for any x G G. This concept was introduced by Foguel. In this note, we call a subgroup H self-conjugate permutable in G if HHX = HXH implies Hx = H. This is the dual of the concept of conjugate permutable subgroups. A Cl-group is a group all of whose cyclic subgroups are self-conjugate permutable; and a C2-group is a group all of whose cyclic subgroups are conjugate permutable or self-conjugate permutable. Some properties of conjugate permutable subgroups and self-conjugate permutable subgroups for a any group are obtained, and the structure of finite Cl and C2-groups are described.

We classify all finite groups G for which Aut(G) is of order 2pg2, where p and q with p > q > 2 are primes, and Aut(G) denotes the full automorphism group of G.

The finite 2-groups G satisfying 'NG(A) : CG(A)' < 2 for all abelian subgroups A of G are determined by the defining relation, and G possesses the class c(G) = l+log2(|G'|).