subject + verb + infinitive

I am having difficulty with subject + verb + infinitive set-ups, as I discovered with who/whom sentences.
I understand who and whom as the subject and object forms.
For sentences that I find a little more difficult I find it challenging to identify the form.

For example:
"This is the man who/whom Sara asked to wait for her."
I figured this would be whom. Since the man is the object of asked; as a variation: "Sara asked him to wait for her." (Subject + verb + object + infinitive + prepositional phrase.)

"This is the man who/whom the authorities refused to permit entry to the building."
I think whom is correct; as a variation: "The authorities refused to permit him entry to the building." (Subject + object + infinitive + object + adverbial phrase?).

I ask because of the following I found online:
"[This is the] man who the authorities refused to admit existed."
I applied the variation: "The authorities refused to admit he existed."
Existed applies to the man and so he is the subject of the verb exist. I'm confused because “he existed” seems to me to be the object in the sentence.

Take for example: “They refused to permit him entry”. The objective pronoun is being used. Both 'to permit' and 'to admit' are transitive here (I think); therefore, isn't “he existed” the object?

I am confused by the use of the infinitive and what qualifies as the object in the subject + verb + infinitive set-up. Any assistance with how to understand this subject + verb + infinitive set-up and the object would be much appreciated.

Answer

Taking your example: "[This is the] man who the authorities refused to admit existed," you have correctly identified that you want to look at the variation "The authorities refused to admit he existed." In that variation "he" is a subject of a clause. "Whom" is used for as a pronoun for the object of the dependent clause, not when the clause itself is the object of the independent clause. So, while "he existed" is the object of the variation, "he" is the subject, and that's what matters.

To take another example:

This is the man whom the doctors refused to admit.

is correct because "the man" is the object of "to admit." In your example, they were not refusing to admit the man, the were refusing to admit a statement about the man.