  The study focuses on strategic-form games extended in the
Eisert-Wilkens-Lewenstein scheme by two unitary operations. Conditions are
determined under which the pair of unitary operators, along with classical
strategies, form a game invariant under isomorphic transformations of the input
classical game. These conditions are then applied to determine these operators,
resulting in five main classes of games satisfying the isomorphism criterion,
and a theorem is proved providing a practical criterion for this isomorphism.
The interdependencies between different classes of extensions are identified,
including limit cases in which one class transforms into another.
