Quasisimple classical groups and their complex group algebras

Let H be a finite quasisimple classical group, i.e., H is perfect and S:= H/Z(H) is a finite simple classical group. We prove that, excluding the open cases when S has a very exceptional Schur multiplier such as PSL₃(4) or PSU₄(3), H is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.