Giving a wave to page, monarch and imperial family (7)

I believe the answer is:

perming

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'giving a wave' is the definition.
Although both the answer and definition are present participle verbs, I don't understand how one could define the other.

'page monarch and imperial family' is the wordplay.
'page' becomes 'p'.
'monarch' becomes 'ER' (abbreviation for Elizabeth Regina, British Queen).
'and' means one lot of letters go next to another.
'imperial family' becomes 'ming' (I can't explain this - if you can you should give a lot more credence to this answer).
'p'+'er'+'ming'='PERMING'

'to' acts as a link.

Can you help me to learn more?

I've seen this clue in The Times.

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