A C♯ Augmented (C♯aug) chord contains the notes C#, A, F. It is built from the augmented formula 1 - 3 - ♯5, with C# as the root.
Is C♯aug the same chord as Faug?
They contain exactly the same three notes — C♯, F and A — so on their own they sound identical. What separates them is context: whichever note the bass and the surrounding progression treat as home becomes the root. Naming it C♯aug rather than Faug is a statement about where the music is, not about the sound of the chord itself.