Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is relevant to some problems of that flavour in various areas of mathematics. I'll lay out the problem I'm interested in. My question, with the natural follow-up: Is there, or might there be, a kind of cohomology relevant to my problem? If so, where should I look, and if not, why not?

The specific question that interests me: Let $Y$ be a mixing (i.e. irreducible, aperiodic) sofic shift in one dimension and let $Z \subset Y$ be a mixing shift of finite type. When do there exist a mixing shift of finite type $X$ and a factor code (= surjective sliding-block = continuous, shift-commuting surjection) $\pi: X \to Y$ admitting a section of $Z$, i.e. a subshift $Z' \subset X$ such that $\pi|_{Z'}: Z' \to Z$ is a conjugacy (shift-commuting homeomorphism)?

In the case $Z=Y$, there are some known obstructions from the work of Mike Boyle, involving periodic points (see e.g. "Lower entropy factors of sofic systems", Ergodic Theory and Dynamical Systems 1984 (4) 541-557). I don't believe the problem has been studied explicitly in the case that $Z$ is a proper subshift of $Y$, where we are asking for $\pi$ to admit a particular local section (i.e. of $Z$), and allowing for the nonexistence of a global section.

I am aware of the use of a kind of cohomology for dynamical systems in, for instance, the work of Bergelson-Tao-Ziegler, but am not sure whether this would be appropriate for problems of the kind I have described.

Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is relevant to some problems of that flavour in various areas of mathematics. I'll lay out the problem I'm interested in. My question, with the natural follow-up: Is there, or might there be, a kind of cohomology relevant to my problem? If so, where should I look, and if not, why not?

The specific question that interests me: Let $Y$ be a mixing (i.e. irreducible, aperiodic) sofic shift in one dimension and let $Z \subset Y$ be a mixing shift of finite type. When do there exist a mixing shift of finite type $X$ and a factor code (= surjective sliding-block = continuous, shift-commuting surjection) $\pi: X \to Y$ admitting a section of $Z$, i.e. a subshift $Z' \subset X$ such that $\pi|_{Z'}: Z' \to Z$ is a conjugacy (shift-commuting homeomorphism)?

In the case $Z=Y$ [Edit in response to Salo's comment: and if we allow $Z = Y$ to be merely mixing sofic], there are some known obstructions from the work of Mike Boyle, involving periodic points (see e.g. "Lower entropy factors of sofic systems", Ergodic Theory and Dynamical Systems 1984 (4) 541-557). I don't believe the problem has been studied explicitly in the case that $Z$ is a proper subshift of $Y$, where we are asking for $\pi$ to admit a particular local section (i.e. of $Z$), and allowing for the nonexistence of a global section.

I am aware of the use of a kind of cohomology for dynamical systems in, for instance, the work of Bergelson-Tao-Ziegler, but am not sure whether this would be appropriate for problems of the kind I have described.