  We prove that Sp\"ath's Character Triple Conjecture holds for every finite
group with respect to maximal defect characters at the prime 2. This is done by
reducing the maximal defect case of the conjecture to the so-called inductive
Alperin-McKay condition whose verification has recently been completed by
Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple
Conjecture for all 2-blocks with abelian defect groups by applying Brauer's
Height Zero Conjecture, a proof of which is now available. We also obtain
similar results for the block-free version of the Character Triple Conjecture
at any prime p.
