Title says it all, but to clarify:

Define a problem, called $ IsInNP$ , as follows:

Given a Turing Machine $ M$ , $ IsInNP$ is the problem of deciding if the problem that $ M$ decides is in $ NP$ .

**What is the complexity class of $ IsInNP$ ? Is it even decidable? Is the answer the same for any other complexity class, like $ NP$ -hard?** And are those questions even sensible to ask?

By the way, I am aware that the class $ NP$ is not enumerable, but since I do not quite understand enumerability and it seems that recursively enumerable problems can be decidable, I do not know if that means that deciding whether a problem is in $ NP$ , or any other complexity class, is decidable.

Also, I am aware of Rice’s Theorem, and I believe it can be interpreted as saying that deciding whether a problem is in $ NP$ is undecidable, but I am not certain.

**Bonus question if the above questions are sensible**: given a property $ S$ that only $ NP$ problems possess, does the above also mean that deciding whether a problem decided by a Turing Machine $ M_2$ has property $ S$ is in the same complexity class as $ IsInNP$ ?