请问这个英语句子的结构该怎么分析？

Question

这句话摘自金融时报：
That begs the question ofprecisely who the policethought they wereprotecting and on whose authority theywere acting.
请问,
who引导的句子，是不是of的宾语从句？
on whose authority在on whose authority they were acting中是不是作状语？正常的顺序是不是they were acting on whose authority.
非常希望您能把句子的结构帮我分析一下

Answer 1

That是主语，begs是谓语，the question是宾语，of后面是同位语性质的定语。即这个定语所说的就是question的内容。大意是：它回避了警方认为他们保护的究竟是谁和他们是奉谁之命采取行动的问题。
who引导的从句和on whose authority可以理解为of的宾语从句，实际上是question的同位语。
on whose authority在on whose authority they were acting中作状语，正常的顺序是they were acting on whose authority，但因是从句，需要倒装。

Answer 2

这个句子成分的确比较特殊！最好先做结构调整才能将其划分的更清楚准确：
原句可以替换为：
Precisely，that begs the question of who the police thought they were protecting and on whose authority they were acting.
=Precisely，that begs the question of whom ( the police thought) they were protecting and whose authority they were acting on .

【Precisely状语】,【that 主语】【begs谓语】【 the question 宾语】【 of 引导介词短语，作宾语的定语】 【（who=whom）（ the police thought 插入语）they were protecting and on whose authority they were acting 宾语从句，说出question 的具体内容】.

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Answer 3

主语·That 谓语Begs 宾语为question 之后of介宾短语结构修饰宾语，‘who’和‘on whose’为两个并列的宾语从句，为of结构的宾语