The 30-year-old, who has also been arrested on suspicion of a malicious communications offence, has been bailed with conditions until September 2.
A police spokesman said the incidents are alleged to have occurred between July 4 and August 4.
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Gosport police posted on Facebook: ‘Off to custody: thanks to some joined-up work from neighbourhoods, response, offender management, town CCTV and the control room, a Priority 1 offender has been arrested and is in the back of the van with Sarge driving him to the Investigation Centre.’