Was the husband's action correct in bringing a maid, then transferring her to the wife's sister due to the wife's objection? And is he obligated to send the maid away and cut off her livelihood to fulfill the wife's desire, or is it permissible for her to remain in Oman under the sponsorship of the wife's sister, knowing that the wife strongly objects to this?

It is not permissible to send the maid back against her will as long as the shortcoming was not from her side, for that contains injustice towards her and deprives her of seeking provision. The wife can be appeased by buying jewelry or gifts. It is important to note that the maid is a non-mahram woman, so private seclusion with her is not permissible, nor is looking at what is not permissible. Also, it is not permissible to burden the worker with what she cannot bear.