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Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform $F^{}$ if $F^{}\otimes T:L_p\left(\right)\otimes X\to L_{p^{\text{'}}}{(}^{\text{'}})\otimes Y$ admits a continuous extension $[F^{},T]:[L_p\left(\right),X]\to [L_{p^{\text{'}}}{(}^{\text{'}}),Y]$. Let $ℱT_p^{}$ denote the collection of such T’s. We show that $ℱT_p^{ℝ\times }=ℱT_p^{\mathbb{Z}\times }=ℱT_p^{\mathbb{Z}ⁿ\times }$ for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then $ℱT_p^{}=ℱT_p^{\mathbb{Z}}$.