HPIL Holdings (HPIL)

[shotsky]

Yes, that would be trading. They could buy up large amounts of shares, but they'd have to hold them to ever have an affect on share price.
A buyback is just that - the company buys shares from other investors, or on the open market, and RETIRES them, reducing the OS. That is a legal act which requires filing with the SEC before it can happen, and it can't happen in a delinquent company.