In your example, under FPTP A gets in.
Under AV it would be decided on second/third/... preferences. Lets assume they don't get exactly the same number of votes - say B does better than C, etc, down to F.
F gets eliminated and the votes shared out. The end result now will be exactly as if F had never stood. Or if you ran another election immediately without F.
Unless A gets over the 50% in the first round, the same thing happens to E, etc, etc.
In answer to your question, YES: it is possible that B will end up winning. B can overtake A if enough of C,D,E,F's supporters put B above A.
Is this fair? I think so. In this situation if A and B went head to head, B would win. So why should A get elected just because a bunch of small parties also stood?