The answer
Ordinarily, transferring crypto between wallets you own is not a disposal for Australian CGT, although a coin amount spent as a transaction fee can have its own tax consequences.
What to do next
Keep a record showing ownership of both wallets, transaction hash, amount, network fee and any relevant valuations.
Koinly crypto tax reports
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Frequently asked questions
Is selling Bitcoin to AUD different from moving it to cold storage?
Yes. A sale is generally a disposal; a same-owner transfer usually is not.
