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We construct a class of non-commutative, non-cocommutative, semisimple Hopfalgebras of dimension $2n^2$ and present conditions to define an inner faithfulaction of these Hopf algebras on quantum polynomial algebras, providing, inthis way, more examples of semisimple Hopf actions which do not factor throughgroup actions. Also, under certain condition, we classify the inner faithfulHopf actions of the Kac-Paljutkin Hopf algebra of dimension $8$, $H_8$, on thequantum plane.

A class of semisimple Hopf algebras acting on quantum polynomial algebras