lifting automorphisms of subgroups of direct products of cyclic p-groups

Let γ be a finite group. a subgroup h of γ is called “fully liftable” in γ if every automorphism of h is the restriction of an automorphism of γ. let g = cpk1 × cpk2 , where 1 ≤ k1 ≤ k2 and p is prime. using information about the subgroup structure of g and knowledge of aut(g), we characterize all fully liftable subgroups of g. it turns out that all cyclic subgroups of g are fully liftable, and non-cyclic subgroups are fully liftable if and only if they are automorphic to certain subproducts of g, where two subgroups h and k are automorphic in g if there exists α ∈ aut(g) such that α(h) = k. further, we compare the fully liftable subgroups of g with the characteristic subgroups of g, which are similarly characterized by certain subproducts. finally, we exhibit some interesting lattice features of both fully liftable subgroups of g and characteristic subgroups of g.