Simple group of component type

Definition
A finite simple non-Abelian group $$G$$ is said to be a simple group of component type if for any involution $$t$$, the group $$C_G(t)/O(C_G(t))$$ has no quasisimple component.

Note that it does not matter which involution $$t$$ we pick because the involutions of a group form a single conjugacy class.