  Essentially generalizing Lie's results, we prove that the contact equivalence
groupoid of a class of (1+1)-dimensional generalized nonlinear Klein-Gordon
equations is the first-order prolongation of its point equivalence groupoid,
and then we carry out the complete group classification of this class. Since it
is normalized, the algebraic method of group classification is naturally
applied here. Using the specific structure of the equivalence group of the
class, we essentially employ the classical Lie theorem on realizations of Lie
algebras by vector fields on the line. This approach allows us to enhance
previous results on Lie symmetries of equations from the class and
substantially simplify the proof. After finding a number of integer
characteristics of cases of Lie-symmetry extensions that are invariant under
action of the equivalence group of the class under study, we exhaustively
describe successive Lie-symmetry extensions within this class.
