Can't pick colors with eyedropper on groups
The title... I can't pick colors with the eyedropper on grouped objects where the color is applied to the group. If I ungroup, the fill and stroke disappears from the object. This is so annoying.

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Mille, I think I know exactly what happens.
Please pick Eyedropper and hit Enter (or double-click the tool’s icon), to open the Eyedropper Options dialog.
I bet you have both Appearance options enabled, both in Picks Up and Applies columns?If yes, please open the Window > Appearance panel to understand what happens when you use Eyedropper to pick and apply anything to a group.
Take a look at the GIF attached.The group has no appearance on tis own, but red-and-orange rectangles inside do.
If you use Eyedropper with both 'Appearance' options unchecked — the tool samples the focal fill and focal stroke from the path clicked and assigns these to paths within the group.
But with the 'Appearance' options checked — the tool samples the appearance as whole (and it’s more than just colors) and assign it to the group — because this is what you had selected. Treat this 'Appearance Eyedropper' as a completely different tool with a very clunky way of switching. It can be very powerful, but is hard to understand unless you keep Appearance panel open (because that section of it Properties shows... well, sucks).The same exact thing happens if you use 'Appearance Eyedropper' with text.
Instead of coloring the characters themselves, the tool would assign a separate fill to the whole object, covering the individual characters fills within.