Spectrum of positive entropy multidimensional dynamical systems with a mixed time

It is shown that if an abelian countable group $G = G_{1}\oplus G_{2}$ is such that $G_{2}$ is a finite group and every aperiodic positive entropy action $\Phi$ of $G_{1}$ on a Lebesgue probability space $(X,\cal B,\mu)$ has a countable Haar spectrum in the subspace $L^{2}_{0}(X,\mu)\ominus L^{2}_{0}(X,\Pi(\Phi),\mu)$, where $\Pi(\Phi)$ denotes the Pinsker $\sigma$-
algebra of $\Phi$, then every aperiodic positive entropy action of $G$ on $(X,\cal B,\mu)$ has the same property. A positive answer to the question of J.P. Thouvenot is obtained as a corollary.