  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.
