I know a little bit about ISI Kolkata since I have a few friends from the institute. Here is what you should know about the internships between semesters. 

The course curriculum specifies that in the summer after completing your first year of M.Tech, you have to do an internship either in the industry or academia. You will have 3 options. 

Industry:

• Many companies that visit the Campus for placements of 2nd-year students also select interns for the same.
• Some of such companies are Samsung Research Labs, Google, General Electric, Reliance, Flipkart, De Shaw, JP Morgan Chase, etc.
• The average stipend offered last year was 65,000 rupees/month.
• Some companies offer more than 1 lakh/month. One of my friends interned at Samsung Research Labs.

Academia:

• As per ISI guidelines, you need to pursue an internship at a different academic Institute or industry.
• A friend of mine went to NUS Singapore, while two others went to a university in Germany. You can also apply to other institutes in Japan, USA, and Europe. People also go to IITs, CMI, and other Tier-1 Institutes of India.

Within ISI:

If for some reason you could not get an internship in the above-mentioned ways, you can approach any of the 250+ faculty members across all 5 ISI‘s and can ask to work with them in their labs.

You will have internships at the end of the first year. In the second year, you will have placements, dissertation, academics. So, the only time left to do an additional internship is the winter break after the first sem. Unless you are extremely enthusiastic, I will suggest that you take a break during this period. Because after the break you will be swamped with coursework for 1.5 years.