To expand on EXTW:
Flexjet is the operator for flights that operate under Part 91 (repositioning flights) and 91K (flights with owners on board). A good portion of the business is now "jet cards" - 25 hour blocks purchased by individuals. These passengers are not owners, but technically charter customers. Here comes the problem: Flexjet is owned by Bombardier - a Canadian company. Since we all know a foreign company can't own a US air carrier (121 or 135), Flexjet contracts with Business Jet Solutions (of which they own 49%) for Part 135 flights. Thus, depending on the situation the same airplane can be "Flexjet" in the morning and "Solutions" in the afternoon.
The biggest difference for the flight crew is the extra line at the beginning of the passenger briefing ("This flight is under the operational control of Business Jet Solutions.") and the different altitudes for oxygen requirements.