There is this book called The Shadow of the Great Game written by an Indian diplomat which talks about what the British stakes were in getting India partitioned during independence.
Nehru, India's first Prime Minister, was not willing to have foreign military presence in India. The area of India close to Afghanistan and USSR was strategically important to the Brits in this constant struggle of their against the USSR. I haven't gotten far enough into the book to know what exactly they did to ensure the partition but this is the story of how Pakistan comes to have such an outsized influence in its internal affairs from Western powers.