  Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the
associated torsion group. In a previous paper, the authors studied, for a given
G, which possible groups G\leq H could appear such that H=E(K)_tors, for
[K:Q]=2. In the present paper, we go further in this study and compute, under
this assumption and for every such G, all the possible situations where G\neq
H. The result is optimal, as we also display examples for every situation we
state as possible. As a consequence, the maximum number of quadratic number
fields K such that E(Q)_tors\neq E(K)_tors is easily obtained.
