This is a question requiring the good knowledge of group theory:
(Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other centralizers NOT isomorphic to these two groups):
i. the elementary group $Z_2^4$, and
ii. the $H_8 \times Z_2$
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(Q2) Which finite groups $G$ ONLY contain specific centralizers isomorphic to these two groups (but contain NO other centralizers isomorphic to anything else):
i. the elementary group $Z_2^4$, and
ii. the $H_8 \times Z_2$
where $H_8$ is the quaternion group with the order of $|H_8|=8$ (be an Hamiltonian). Here $Z_2$ is the cyclic group of the order $|Z_2|=2$ .
Let us consider the $|G|$ be as small as possible. Your answer only needs to provide just AN example, the order of the group G and its all centralizers. (NO need to be complete.) :o)
see also this.
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.