By the famous result of Brauer and Fowler, there exist finitely many simple groups with given involution centralizer. There are many results which determine all finite simple groups with given involution centralizer. Is a complete solution known?

That is, do we have a complete classification of

- Groups $H$ such that $H \cong C_G(t)$ for some involution $t$ of a finite simple group $G$?
- For each $H$, a list of all finite simple groups $G$ such that $C_G(t) \cong H$ for some involution $t \in G$?