I'm pretty certain I am in fact fingering a G as I wasn't using my pinky at all, and yes the G2 is coming out a half-tone higher than it should be doing. The G isn't quite as bad, but still not great.
It's easy enough to check the action of the G# mechanism.
On a modern horn it'll be a two=part mechanism; there's the lever key (the bit you press down) arrowed in red and the cup key, arrowed in blue.
A bar (or key arm) on the lever key sits over a small stub on the cup key and prevents it from rising unless the lever key is pressed down.
In its resting state the lever key is sprung upwards so that the touchpiece lies level with the bell key touchpieces and the bar on the other end is pressing down on the cup key stub. The cup key is sprung so that it always wants to open. As long as that bar is fully down over the cup key stub, it can't rise.
So, if it's rising when it shouldn't do then it means the lever arm isn't pressing down on the stub fully - and there may be several reasons for this.
Press the G# lever key down and release it. Do you feel any resistance when you press it down? When you release it, does it rise smartly until it's level with the rest of the bell keys? If not then it's likely that the spring that powers the key is at fault (broken or dislodged). This is an easy fix.
A good way to confirm this is simply to press down on the lever key arm....if it moves down and closes the cup key, it's definitely a springing problem.
However if the lever key rises smartly to be in line with the bell keys and the cup key is still not closed, it's quite likely that either the G# lever key has got bent (best case scenario) or that the entire bell key group is out of alignment...and this is usually because the bell has been knocked out of line (worst case scenario).