[Grammar] What have you been doing with yourself?

[sitifan]

What have you been doing with yourself? (Practical English Usage, by Michael Swan, section 458.2)
https://www.phrasemix.com/phrases/what-to-do-with-oneself
In the above sentence, is the pronoun "what" an object of "doing" or an object of "with"?

 

[jutfrank]

If you rephrase the question (unnaturally, I might add) in indicative form, it becomes clear:

You've been doing what with yourself?