Is it permissible to record a teacher's lesson without her permission to help a sick friend?

It is permissible to ask the professor for permission to record the lecture for a sick colleague, and it is expected that he will grant permission. If he does not grant permission, then the lecture should not be recorded. You can help your colleague by conveying to her what you have learned from the professor, for in that there is a reward for teaching and safety from the issue of the professor's rights.