I was an F-1 student in the U.S. for more than 5 years and filed taxes as a resident alien.
The employer, for whom I worked on CPT last summer, did not withdraw SS/Medicare taxes from my salary. I was informed about it during the summer, but didn't realize that these taxes had to be withdrawn (I realized it only recently while preparing my tax forms). I contacted my employer, but apparently, it takes them some time to look at these things.
As I understand, I will need to file form 8919 and pay my share of unpaid SS/Medicare taxes by April 15.
For personal reasons (travel abroad), I am planning to file for an automatic 6-month extension for filing federal tax return. Will it also extend the deadline for SS/Medicare tax payment?
I expect my federal tax return to be ~$3000, while my unpaid SS/Medicare is ~$1500.
Do I need to do any payment by April 15 to avoid interest and fees if I file for an federal extension?

You do not file form 8919. You can not pay FICA taxes yourself. It must be through your employer.
So file your tax return without any change on your tax return.