Here is the answer to your query-
The first ‘I’ in IIIT Hyderabad stands for ‘International’. It is a private ICT Research University that was founded in 1998. The first ‘I’ in IIIT Delhi stands for ‘Indrasprastha’. It is a government institute and comes under the Delhi Government. The first ‘I’ of IIIT Allahabad stands for ‘Indian’. IIITA and IIIT Pune come under the collective of institutes that are actually the IIITs, as in, the Indian Institutes of Information Technology.
IIIT Allahabad is one of the 5 IIITs that were formed by MoE, and it has the tag ‘Institute of National Importance’. IIIT Pune is among the other 20 IIITs that were formed under the PPP (Public Private Partnership) model. It is also an Institute of National Importance.
IIIT Hyderabad is one of the best in India for CS. The CS department of IIITH boasts of top-notch placements, a good number of entrants into GSoC, and high participation rates in Competitive Programming.
Only IIIT Allahabad and IIIT Pune belong to the collective of Indian Institutes of Information Technology. The other institutions mentioned by you are different from this group.
IIIT Delhi is an autonomous college (of Delhi Government), while IIIT Pune is affiliated to Pune University. IIIT Hyderabad is a private autonomous college.
The rest of the 18 IIITs are set up on a public-private partnership model. They are funded by the central government, the respective state governments, and industry partners in the ratio 50:35:15. IIIT Allahabad is the only one in this list that belongs to this group.
According to the IIIT Act, 5 of these 18 IIITs have been declared as the ‘Institutes of National Importance’. These are- IIIT Allahabad, IIIT Gwalior, IIIT Jabalpur, IIIT Kurnool and IIIT Kancheepuram.
Thus, none of the colleges mentioned by you are related to one another.