Do I have to serve notice period if I resign during extended probation period?

In the appointment letter, there is no separate mention of notice period during probation. However, under notice period section, it states "your employment may be terminated at any time by two month's notice in writing by either side or by paying you two month's consolidated salary in lieu of the notice. In case you leave our employment without notice, we shall have the right to deduct as liquidated damages an amount equivalent to 2 months' salary from any monies that may be due to you." Now I dont want to serve notice period, but the HR says I have to. It doesn't matter if I am under probation or not. What is the rule so I can fight my case

Since your appointment letter specifies notice period of two months and there is no specific mention of the notice period during probation, you are bound by the two month notice, but normally during the probation there is lesser notice period than is applicable to a permanent employee. But I have not come across any case law or definite rule to that effect.

You are bound by the agreement of 2 months notice
Whether you are on extended probation or not

The company put a new recruit on the probation period in views that whether this new employee is able to cope up with the requirements of the job or not. If company finds things to be satisfactory and find the employee to be suitable for the job, company confirms the job of the employee and he gets the benefits of confirmed employee.
Whether in your appointment letter, the clause to extend the probation period is exist or not ? Whether anything in writing the probation period has extended ? If it is no, then you are not come within the ambit of notice, since you have not yet confirmed to employment.
Dear friend, there is nothing clear rather it is interpretation base. We keep probationer from the ambit of notice till he confirmed. You can inform your employer that since my job has not confirmed as yet, I am in deep mental stress for unforeseen happenings. therefore,I consider to leave the job still under uncertainty under umbrella of Probation which is never ending.
Still I advise you to contact a good lawyer dealing on subject of labour law.

Usually the notice period will be half when compared to the regular employee at that standards of job role.