This English phrase - also in the singular, "regulatory authority" - is the best functionally equivalent translation. The adjective "regulatory" is synonymous with "supervisory" Americans tend to use the word "agency" in lieu of "authority; such as "regulatory agency" or "government agency"; both also adequate translation of "Aufsichtsbehörde" for an American audience.
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public bodies that make and enforce relevant rules and regulations in relation to, for example, a specific industry
The relevant regulatory authorities should be informed of the company's recent activities.