There are loads of travel agencies that plan such treks..we went to one last year..to hemkund and valley of flowers..You can even check india hikes...they arrange treks and most of them are group treks!!
If you want to do treks in South India Kumaraparvatha trek is the biggest one so far I know. You can contact Bangalore trekking club for trek with strangers. They are arranging weekend treks. If you want to do treks in the North you can treks like Hampta pass trek, pin Parvathi valley trek, Kuari Pass trek.
Depends on the place you want to try. There are many trek companies in both north and south india conducting such activities. I am not sure if giving third party commercial links is allowed here or not. But you can always google search (Place name) + trek + Fixed departures and you'll find results.
Hi, I have attached with this comment few trekking options for you , you can click on them to find more details about them. Also if you want to book any of these, you can simply fill in your details in the form you get after clicking on the link.