Let $\pi : X \to Y$ be a factor map between subshifts over finite alphabets. Let $\operatorname{Aut}(X)$ and $\operatorname{Aut}(Y)$ stand for automorphism groups of these shifts. We say that $\varphi \in\operatorname{Aut}(X)$ is compatible with $\pi$ if $\pi(x)=\pi(x')$ for $x,x'\in X$ implies $\pi(\varphi(x))=\pi(\varphi(x'))$. If $\varphi$ is compatible with $\pi$ then one can define $\psi=\rho_\pi(\varphi)$ on $y\in Y$ by taking $x\in X$ with $\pi(x)=y$ and setting $\psi(\pi(x)) = \pi(\varphi(x))$. Clearly, $\psi$ is an endomorphism of $Y$. Consider the following conditions
$\psi=\rho_\pi(\varphi)$ is an automorphism of $Y$ for some $\pi$-compatible automorphism $\varphi$ of $X$,
every automorphism $\psi$ of $Y$ satisfies $\psi=\rho_\pi(\varphi)$ for some automorphism $\varphi$ of $X$ compatible with $\pi$,
for every $\pi$-compatible automorphism $\varphi$ of $X$ the endomorphism $\psi=\rho_\pi(\varphi)$ is an automorphism of $Y$,
every automorphism of $X$ is $\pi$-compatible.
Are there any conditions one can impose on $X$ and $Y$ to guarantee that conditions (1)–(4) or conditions from some subset of $\{1,2,3,4\}$ hold?
