What British group gained control over India after the collapse of the Mughal Empire?
The group that you are asking about is the British East India Company. This company came to dominate India after the fall of the Mughals and continued to rule until the British government took control of India in a more formal way in the 1850s.
When Britain first got involved in India, it did so through the British East India Company. It did this rather than trying to actually send government officials out to India. The Company became more and more involved in the politics of India. Eventually, the Mughal Empire declined and Company power increased. This came to a head at the Battle of Plassey where Company troops led by “Clive of India” defeated the last independent nawab of Bengal and took a position of dominance for the Company.
The group, then, is the British East India Company.