Hi Good Morning

As a complement to the answer of our fellow attorney, and according to the Qatar Family Law No. (22) for the year 2006 regulating personal status matters, the matter is regulated by a law that cannot be bypassed, as the law stipulates in Article (173) :Custody of children granted to women shall terminates when a male child completes thirteen (13) years of age and the female child completes fifteen (15) years of age, unless the Court rules
otherwise. After investigating the interest of the Child, the Court may permit such custody to continue until the male Child attains fifteen (15) Years of age and the female Child consummates
marriage, or enable Child under custody to choose between the disputing custodians after investigating their suitability. In all cases, the Court shall give the reasons for its decision. Notwithstanding the provisions of the preceding paragraph, female custodianship may continue if the Child under custody is mentally ill or his illness is debilitating. The agnate or guardian may not take the Child after termination of the female custodianship except by consent or judicial permission.
If there is no judicial decree on guardianship and the agnate or guardian forcibly took the Child during the age of female custodianship, the Judge may, upon request of the female entitled, temporarily return the Child to her, with or without her personal surety, and direct the agnate or guardian to file a guardianship claim before the competent Court.
Article 176:
The guardian of the Child may retain the passport of the Child save for when travelling when it shall be delivered to the female custodian of the Child. The Judge may order that the female custodian shall keep the passport of the Child if he considers
that the guardian may not hand over the passport timely when needed. The female custodian may keep the original birth certificate of the Child and any other evidential documents of the Child, or certified copies thereof in addition to his identity card.
Article 177:
If the divorced female custodian is a foreigner living with her relatives in Qatar or resident under the sponsorship of another before marrying, her divorcer shall transfer her sponsorship to another suitable sponsor. If he refuses, the Court shall order the transfer of her sponsorship. The guardian may not cancel the sponsorship of the Child under custody until expiration of the custody.
** Therfore, if the mother wishes to travel, the child may be prevented from traveling and asked to drop custody of the mother by filing a lawsuit at the court and requesting that the custody be taken away from her with a request to prevent travel of the child for the purpose of staying in her father’s residence place.

The approval of the employer must be obtained for the purpose of the mother's travel and the extension of the visa before departure, otherwise it is not possible for her to return. The business authority can impose a ban on her and the inability to return, so it is better to obtain the approval of the company and extend the visa and we do not recommend leaving in this way