I'm assuming that the G-string is an un-wound type, apparently that's why the G-String always causes a lot of grief with regards to the tuning on electric guitars, back in the early days of electric guitars, sets of strings came with wound G-Strings and plain B and high E-strings, it was only when guitarists like Jimi Hendrix started using un-wound strings for the G-string on electric guitars to extend the range for soloing that string manufacturers started making sets of strings with an u-wound G-string, the un-wound G-string on my Gold Strat is pretty hard to keep in tune, you only have to slightly tweak the tuner to make the G-string go a bit sharp or flat by at least 5 cents, even though the break-angle over the nut is fairly shallow (about 2-3 degrees from horizontal).

The string could be binding in the nut slot a bit.