I have a 1 year mutilple entry tourist visa for India which allows me a maximum stay of 90 days. About 1 month after I reached India, I have secured a 3 month training at PGI. Now the total duration of my stay here will be around 4.5 months. How I can stay for an extended period and what will be the procedure to extend my visa to enable me to complete my training course?
You have a tourist visa, the standard 180-day visa with multiple entry, and participating in a training program is not allowed on that visa. That would require issuance of a new visa, for which you would have to exit and reenter the country.
India is extremely stringent with visa extensions. The rare exceptions are those granted to individuals of Indian origin, or those who encounter an emergency (medical, stolen passport and, in it, the visa needed to exit). Changing the purpose of your stay in India would not be considered a valid reason.
You can ask about a 14-day extension at the Ministry of Home Affairs in Delhi, 26 Jaisalmer House, Man Singh Rd, 011-23385748 firstname.lastname@example.org
Another alternative is the Foreigners’ Regional Registration Office, Level 2, East Block 8, Sector 1, Rama Krishna Puram, Delhi, 011-26195530 email@example.com
While there are FRRO regional offices, it appears that smaller, local authorities are much less likely to entertain such requests.
Edit: @DJClayworth contributed an important note:
Just to point out to OP what I hope is obvious, if you have already started your training with your current visa, you are breaking the law.