Here are some links for internship programmes at various research institutions (in no particular order.)
The applications for these programmes open at different times, so keep checking these websites for the latest info. Some of these programmes are open to students from all countries and some are exclusive to indian students. Some of these programmes take students from IITs only.
When applying to any of these programmes, try to apply through the Training and Placement Cell – your application will get more attention if it comes through the TnP Cell.
One very important thing to note is – DO NOT spam professors with unsolicited applications for internships. Those are considered spam and are not answered. The only thing they do is harm the reputation of your institute. So refrain from sending mails titled “IIT student seeks internship at your esteemed institute“, no one even reads them. If you are genuinely interested in somebody’s work, only then send them a request, but make sure to include some work you have done related to their work, tell them why it would be beneficial for them to work with you, etc. If your friend can tell from your email that you are lying, they can tell as well.
Another thing to note is that if your grade point is less than 8.0, don’t bother applying for these programmes, you will not get selected. Most of the students who get selected for these programmes have a very high grade point (generally more than 9.0) There are a lot of other places where you can do an internship.
One final thing I want to say is – Do not step on each other’s toes. If you have a good internship be content with it.