Almost quasisimple group

Definition
A group is termed an almost quasisimple group if it has a self-centralizing normal subgroup that is a quasisimple group.

Note that for such a group, the self-centralizing quasisimple normal subgroup is unique (and hence also a characteristic subgroup) and is also the only component of the group, hence also equals the commuting product and the generalized Fitting subgroup of the group.