ANOBAG does not apply to employers not in the EU and there is no obligation for your husband's company to register in Switzerland.

You need to speak to the AVS/AHV. It seems OTT for your husband to go through a payroll company as the US company has no AVS/AHV obligations in Switzerland.

The permit needs to be discussed with the migration authorities responsible. He does not complete with EU/Swiss employees. He needs to show he is a professional and there isn't salary dumping going, etc(if he wants to be employed in Switzerland). To work for a foreign company I don't know the permit implications. Migrations would. Call the canton migrations office.

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