Hi Matija,
In the case of triads with a sharp or flat root note it’s always worth going back to the version with the natural root note. In the case of C# aug, you could check C aug at first: C E G#. Then, C# aug will have all 3 notes sharpened by a semitone: C# E# G##. Although this is enharmonically equivalent to “C# F A”, the convention is to use each note name (“letter”) once, and these note names come from the major scale of the root note.
The notes of the C# major scale are:
C# D# E# F# G# A# B# (all of them a semitone higher than the notes of the C major scale)
In an augmented triad, you have the root (C#), the major 3rd (E#) and the #5 (G##). As the rule says, you should not repeat letters, use different accidentals (flat/sharp) or use enharmonic equivalents. This is why C# F A is not a correct answer here.
Also, in these exercises, it is always worth recalling the basic triads you can form in C major:
CEG
DFA
EGB
and so on.
In the case of G B Eb, you can see that it will be some sort of Eb triad. Let’s see E major first:
E G# B
Thus, Eb major will be: Eb G Bb
So, Eb G B is an Eb aug triad.
But your observation of augmented triads is also correct. Augmented triads consist of a major third stacked on another major third. In the note circle an augmented triad looks like this (sorry for the rudimentary illustration):
The augmented triad is unique in that all 3 notes are the same distance (interval) from each other. It means that the same 3 notes can be either the root, the major 3rd or the #5 of the “same” augmented triad, “same” here meaning that the chord would sound the same when played. But, when named properly, it would be something like this (using C aug as the starting point):
Aug triad 1: C E G#
Aug triad 2: E G# B#
Aug triad 3: G# B# D##