“For precisely what federal crimes is the president of the United States under investigation by a special counsel appointed by the Justice Department?”
The investigation is not necessarily limited to actions by the President, but also includes those in his campaign orbit. Could it include the President? Maybe. But the Special Counsel was appointed for an independent investigation, because Trump obstructed justice by firing the person leading the original investigation, James Comey.
https://www.nationalreview.com/2018/09/trump-russia-probe-robert-mueller-investigation/