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For the Klein group K and a positive integer k, irreducible modules of the orbifold vertex operator algebra L픰픩2 ̂(k, 0)K have been classified and constructed in [C. Jiang and B. Wang, Representations of the orbifold VOAS L픰픩2 ̂(k, 0)K and the commutant VOAS CL픰픬 m ̂(1,0)⊗3(L픰픬m ̂(3, 0)), Bull. Malays. Math. Sci. Soc. 44(1) (2021) 353–373]. In this paper, we determine completely the fusion rules of L픰픩2 ̂(k, 0)K. [ABSTRACT FROM AUTHOR]