Soficity of free extensions of effective subshifts

Let $G$ be a group and $H\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there are always $H$-subshifts which are effectively closed by patterns whose free extension to $G$ is non-sofic. We also present a few applications in the form of a new simulation theorem and a new class of groups which admit strongly aperiodic SFTs.