what

CJ 

I work this out by substituting other indirect questions like whether It is permitted.  If a large range of indirect questions doesn't work, I suspect that only that which is the correct interpretation.

a description of whether it is permitted doesn't work for my ear, so I think that which is probably a better characterization in that case.

learn whether it is permitted does work for me, as well as learn that which is permitted, so there I see a greater possibility for ambiguity. 

The verbs of knowing, finding out, and learning tend to be followed by indirect questions, so when these are followed by what, where, whether, how, etc., I am suspicious that an indirect question is a possible interpretation, along with the that which interpretation, thereby creating the ambiguity you called attention to in your original post.

CJ 

Many of these grammatical terms, like 'interrogative', are not as tightly defined as the terms used in mathematics, so it is sometimes hopeless to try to disentangle the meanings with mathematical precision.

I don't think there's any problem if you want to say that both of these have interrogative aspects.

CJ 

It would be much easier to come up with additions.

ask, inquire, wonder, guess, know, learn, find out, decide, and any cases with the infinitive after the wh- word are typically the interrogative case.  Also in this category are many negative expressions of not knowing:  I can't imagine, I have no idea, ...

We learned how to read Japanese.

Do you have any idea what she said?

I can guess very easily who was there. 

Have you found out who signed the document?

Jane asked what to do.

I wonder if Ken is going to be there.

I don't know whether to go or to stay.

No one could decide what to do. 

_____ 

In my opinion, there are many ambiguous cases.

CJ