The two private equity firms each received $30 million for structuring the original 2006 deal and contracted to receive additional fees for providing merger advice to the retailer after the acquisition. Had the firms been registered and regulated as broker-dealers at the time of the buyout, that advice and fee might have been examined for potential conflicts of interest. The management agreement negotiated with Michaels also entitled the two firms to share fees of $4.7 million for the I.P.O. last month.

At the time of the buyout, Blackstone was preparing to become a public company itself, and the next year it registered its capital markets unit as a brokerage firm. Bain, on the other hand, remains unregistered, even though it has earned hundreds of millions of dollars in fees generated by activities that appear to require broker-dealer regulation.

A spokesman for Bain declined to comment.

For decades, the private equity industry was almost completely unregulated. Private equity firms use borrowed money to set up partnerships that acquire companies that they hope to resell later at a profit. The firms sell interests in the partnerships to endowments, pension funds and wealthy individual investors. In 2010, Congress stepped in with the Dodd-Frank Act, which required private equity firms with more than $150 million in assets to register in the category of investment advisers. That registration process began in 2012.

As regulatory regimes go, however, oversight of broker-dealers is much stricter than it is for investment advisers. Brokers receive more frequent audits and examinations — only 9 percent of investment advisers are examined annually, compared with 55 percent of broker-dealers — and face capital requirements and greater legal liabilities.

The American private equity industry, which manages $3.5 trillion in assets, wants less regulation, not more. Steve Judge, president of the Private Equity Growth Capital Council, the industry trade group, characterizes the financial work provided by these firms as investment advisory services. “Layering broker-dealer regulations on private equity will be of no meaningful benefit to investors and would levy significant costs on private equity firms,” he said in an emailed statement. “We will continue to work with the S.E.C. to ensure that private equity investment advisers are not inappropriately miscategorized as broker-dealers in the future.”

Yet there is a compelling argument that many if not all such firms are, in fact, broker-dealers. The Securities Exchange Act of 1934 states that “any person engaged in the business of affecting transactions in securities for the account of others” should register as a broker and submit to heightened oversight intended to ensure that its customers are treated fairly.

Image Andrew J. Ceresney, the S.E.C.’s chief of enforcement, said he had nothing to do with who was subject to broker-dealer oversight, adding, “Once they make the rules, I enforce them.” Credit... Gary Cameron/Reuters

S.E.C. guidelines say much the same: Firms may have to register as broker-dealers if they participate in “important parts of a securities transaction, including solicitation, negotiation or execution of the transaction” and if compensation or participation in the deal depends on the outcome or size of the transaction.