The following may kindly be noted.
If it was a recording mistake of the HR (or personnel department) that, despite the employee having submitted a proper Transfer Certificate (TC), they recorded it wrongly, then the employee definitely gets a chance for an extension.
If the employee had given other records showing a different date of birth while joining the service and now is claiming a different date as per the TC, then there is no chance for an extension.
In the case of many public sector companies, the high courts/Supreme Court have held that a dispute regarding the date of birth at the fag end of the service period is not accepted. Citations for these could be obtained from the e-version of labor law journals, or any local lawyer can help.
However, it is also the responsibility of the employee to ensure the proper recording of their date of birth in the organization, which is possible through verifying various service records. If the organization has no system of any such records, then at least when the employees obtain service/working certificates for applying for loans, etc., the date of birth/remaining years of service mentioned can be verified clearly.
Also, many organizations have a system of indicating the date of birth and age of retirement in the appointment letter itself at the beginning.
If, according to Vasukumar, nothing of the above has happened, then there is no way the employee can ask for an extension now.
Kind regards,
Dayanand L Guddin