I am struggling to understand the notion of quadratic twists for elliptic surfaces. For elliptic surfaces, the singular fibres are classified by Kodaira's classification. A quadratic twist of an elliptic surface $S$ acts on the singular fibres as a "transfer" of the star, i.e. $I_n^* \leftrightarrow I_n$, $II\leftrightarrow IV^*$ etc. Can such operations be understood as an automorphism of $S$, or perhaps a quotient by an automorphism of $S$?
Ideas: It seems that quadratic twists are essentially 1-isogenies. Are such quadratic twists therefore encoded in the Mordell-Weil group of $S$?