AIB had yesterday tweeted a picture of a Modi look-alike peering into his mobile phone at a railway station alongside an actual photograph of the PM, with props superimposed with the help of Snapchat's dog filter, an app.
The dog filter is a special effect application which allows users to add a dog's nose, ears and tongue over their faces while taking a selfie or on other photographs.
The post had a hashtag that apparently referred to Modi's frequent foreign visits.
The police forwarded his tweet to the Cyber Police Station, which registered an FIR.
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"After legal consultations, an offence was registered against AIB and a further probe is on," said Mumbai police spokesperson DCP Rashmi Karandikar.
DCP Cyber Crime Akbar Pathan said a case has been registered against AIB on charges of "defamation" and "publishing/transmitting obscene material in electronic form".
Text messages and phone calls to AIB co-founder Tanmay Bhat and Rohan Joshi, who is a key member of the group, went unanswered.
After the post came under fire from Twiterrati, Bhat had yesterday posted a screenshot of what seemed to be an old tweet by Modi which said "we surely need more humour in public life".