Which non-Abelian finite groups contain the two specific centralizers? - part II
Solution 1:
The comments suggest that you mean only centralizers of involutions. Even in that case, no finite group $G$ can have only the two involutionn centralizers you suggest, so Q2 seems to have a negative (or empty) answer. One involution centralizer must contain a Sylow $2$-subgroup. Hence the Sylow $2$-subgroup of $G$ must have order $16$. But the elementary group of order $16$ and the $H_{8} \times Z_{2}$ are then both Sylow $2$-subgroups of $G$, a contradiction, as they are clearly not conjugate.