Caused to continue with a pet reputed to go out (11)

I believe the answer is:

perpetuated

'caused to continue' is the definition.
('perpetuate' can be a synonym of 'cause to continue')

'a pet reputed to go out' is the wordplay.
'to go out' indicates an anagram.
'a'+'pet'+'reputed'='apetreputed'
'apetreputed' is an anagram of 'PERPETUATED'.

'with' is the link.

I've seen this clue in The Irish Independent.

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