  Let $\Gamma$ be a countably infinite group. Given $k \in \mathbb{N}$, we use
$\mathrm{Free}(k^\Gamma)$ to denote the free part of the Bernoulli shift action
of $\Gamma$ on $k^\Gamma$. Seward and Tucker-Drob showed that there exists a
free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^\Gamma)$ such that every
free Borel action of $\Gamma$ on a Polish space admits a Borel
$\Gamma$-equivariant map to $\mathcal{S}$. Here we generalize this result as
follows. Let $\mathcal{S}$ be a subshift of finite type (for example,
$\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of
$\Gamma$ with some finite number of colors). Suppose that $\pi \colon
\mathrm{Free}(k^\Gamma) \to \mathcal{S}$ is a continuous $\Gamma$-equivariant
map and let $\mathrm{Stab}(\pi)$ be the set of all group elements that fix
every point in the image of $\pi$. Unless $\pi$ is constant,
$\mathrm{Stab}(\pi)$ is a finite normal subgroup of $\Gamma$. We prove that
there exists a subshift $\mathcal{S}' \subseteq \mathcal{S}$ such that the
stabilizer of every point in $\mathcal{S}'$ is $\mathrm{Stab}(\pi)$ and every
free Borel action of $\Gamma$ on a Polish space admits a Borel
$\Gamma$-equivariant map to $\mathcal{S}'$. In particular, if the shift action
of $\Gamma$ on the image of $\pi$ is faithful (i.e., if $\mathrm{Stab}(\pi)$ is
trivial), then the subshift $\mathcal{S}'$ is free. As an application of this
general result, we deduce that if $F$ is a finite symmetric subset of $\Gamma
\setminus \{\mathbf{1}\}$ of size $|F| = d \geq 1$ and $\mathrm{Col}(F, d + 1)
\subseteq (d+1)^\Gamma$ is the set of all proper $(d+1)$-colorings of the
Cayley graph of $\Gamma$ corresponding to $F$, then there is a free subshift
$\mathcal{S} \subseteq \mathrm{Col}(F, d+1)$ such that every free Borel action
of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to
$\mathcal{S}$.
