The troubled for-profit education company that owns the giant University of Phoenix agreed on Monday to be bought for $1.1 billion by a group of investors that includes a private equity firm with close ties to the Obama administration.

The university and its owner, the Apollo Education Group, have been subject to a series of state and federal investigations into allegations of shady recruiting, deceptive advertising and questionable financial aid practices.

In recent years, many for-profit educational institutions that have received billions of dollars in federal aid, including the University of Phoenix, have been pummeled by criticisms that they preyed upon veterans and low-income students, saddling them with outsize student loan debt and subpar instruction.

Moreover, at many of these schools, enrollment has been falling and profits shrinking, casting doubt on the future health of the industry.