Steps to reproduce:
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make a fresh intall of Firefox with Avast installed
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log in to your Firefox sync account with your email and password
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you can wait or force a manual sync
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nothing will happen
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you can check errors at: about:sync-log - especially “TokenServerClientNetworkError”
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close Firefox
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turn off Avast for 10 minutes
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start Firefox
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choose manual sync
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it will sync all your bookmarks, extensions, etc.
Tested with newest Avast, Win 10 64bit, Firefox 41 beta