Questions & Answers about Она попросила разрешения.
Because it is past tense feminine singular. In Russian, past-tense verbs agree with the subject in gender and number:
- он попросил
- она попросила
- оно попросило
- они попросили
Since the subject here is она, the verb must be попросила.
The dictionary form is попросить. It is a perfective verb, so it presents the action as a completed single request.
Its imperfective partner is просить:
- Она попросила разрешения = she made the request / asked once
- Она просила разрешения = she was asking, used to ask, or asked repeatedly
In English, both may simply be translated as asked, so this is an important Russian aspect difference to notice.