Interesting harmonies can often be produced by moving a single chord shape/type around (transposing it), rather than by strictly using chords within a particular mode, scale or key. One example that has always fascinated me, for instance, is transposing major chords by the intervals in a minor pentatonic scale, something that is often done in pop/rock music. As an example, the chords E, G, A, B and D contain altogether the pitches E, F#, G, G#, A, B, C#, D and D# - quite a complex pitch set (or scale if you like), produced in such a simple way, yet with a "sound" that most listener's ears will be used to.
However, your example is not as complex as this.
If your chord sequence feels like it emphasises the A chord (i.e. it comes "to rest" on this chord), then I would say you are correct that this is essentially VI-VII-I in the key of A Minor (using the Aeolian, the Natural Minor), but with the tonic chord (I) chromatically altered to be major (i.e. the third is raised). You could alternatively view this as being in A Major, but with chords borrowed from A Minor, but this seems less convincing to me.
Certainly playing the sequence F-G-Am sounds essentially the same as your chord sequence, except that yours has the little "surprise" given by the unexpected major I chord.
In terms of soloing over this, I guess it depends upon style. A Minor Pentatonic and A Aeolian obviously work over the first two chords, but in more "bluesy" popular styles it is common to use the minor third over a major chord, so you could use either of these scales (which both have C natural, the minor third) for the A Major chord, too.
@Pat Muchmore's answer is pretty convincing. So, having read it, I played this chord sequence a few more times. The "feel" of each chord seems dependent upon context to me. If I strum a few bars of Am then play F-G-A, the F and G chords seem entirely natural, but the A chord seems unexpected - no surprise here though(!), in this case we expect the F and G to be VI and VII in A Minor. If I strum a few bars of A before playing F-G-A, the F chord sounds most "alien" and A Major feels like home.
In the second case, where A Major is the home key, the G-A sounds essentially like a Mixolydian relationship (bVII-I). And so in this case, the F chord is the unexpected chord. In fact, if A is established as the home tonic chord, this chord sequence has a compelling "inevitability" about it: F feels outside the home key (high tension); G really wants to move up to the home chord (different kind of tension); A we're here (release)!
Whichever way you hear this chord sequence, it will always have some sense of ambiguity as the chords cannot all be in one diatonic key. But, I would say the function (and so "feel") of each chord, is dependent upon whether A Major or A Minor have already been established.
In either case, A Minor Pentatonic works well over the chord sequence, for the reasons already given above. However, A Aeolian doesn't seem appropriate over the A Major chord, but fits nicely over the first two chords.