The strong independence theorem for automorphism groups and congruence lattices of arbitrary lattices (Q1586920)

The first of the two authors raised the following problem: Let \(L\) be a nontrivial lattice and let \(G\) be a group. Does there exist a lattice \(K\) such that \(K\) and \(L\) have isomorphic congruence lattices and the automorphism group of \(K\) is isomorphic to \(G\)? Up to now, partial solutions of this problem have been known. The authors give an affirmative solution of this problem (actually, they prove a stronger result). Also, they offer new problems.