The A C D A by it could be in the key of A but there is no C major in the key of A.
Billy, pick up your guitar and play this progression.
C for 4 beats
Eb for 4 beats
F for 4 beats
Ab for 2 beats
Bb for 2 beats
Repeat a few times.
You are playing major chords with the notes of a C Aeolian (minor) scale as the roots.
There is no C in the key of A
MAJOR. The keys of A have 7 modes. 4 of the A modes have a C natural in it. They just aren't the major flavor scales. Here's a chart. With an A as the root, 2,3,6 and 7 have a C natural.