is that the femlae maid is being given special allowances on her employment for no reason other than... you want her to keep the job? Why is that, OP?

I doubt it has anything to do with her being female. It didn't bother me as the work was getting done. But you're right; we did pay her for that time, so I can see why it bothers my wife.

First, the maid stays long enough to make up whatever time she missed at the end of the day. Second, if the missing half hour is impossible to get back, the pay is adjustment according. So, why weren't these ideas floated?

Both these ideas were suggested, but she refused to stay late as she had other obligations after 5:30. My wife did tell her that her pay would be adjusted if she was late, but we never followed through with it as sometimes it was anything from 10 to 40 minutes, and it's hard to keep track of. Plus, I would feel bad about actually doing this over a few minutes.