Although it could be a mechanical problem, it sounds more likely to be carburation.
One of the challenges with modern motorcycle engines equipped with fuel injection is that faults can sometimes be difficult to diagnose. This is because the bike's brain (the ECU or BMS) relies on a number of sensors to allow it to achieve the correct air-fuel mixture. There are two sensors on the throttle body, a temperature sensor and an O2 sensor in the exhaust downpipe. If any of these are the cause of the problem it is better to start by checking whether the ECU is reporting any fault codes. Unfortunately for the F650GS this requires access to a GS-911 code reader. Checking these sensors without a GS-911 reader is possible but much more difficult and sometimes, for example in the case of the O2 sensor, the only way to check them is to replace by a known good one.
2006 F650GS in Black