Plausibility Check: India not in the Commonwealth after its independence?

Since I have not been able to find a lot of information about this on the Internet, I turn to my fellow AH.com members to obtain an answer for this question.

What factors led India to join the Commonwealth OTL after its independence, was membership offered by the United Kingdom, or did the Indian government explicitely asked for Commonwealth membership, if yes why?

I would be very interested to know if in the eyes of the members, a scenario where in British eyes Indian is just another foreign country once it becomes independent is plausible and how it could happen.

To be such a scenario does not seem impossible considering the highly symbolical nature of the Commonwealth and the fact that the benefits Britain grants to Commonwealth subjects are reciprocated by no other member. Britain calling the whole this off using the argument "you are now independent, deal with the consequences!" could in my opinion happen in the right circumstances.

Anyhow, I would be glad to get opinions on this.
 
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