Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a solvable group of automorphisms permutes the involutions transitively. The Suzuki 2-groups were classified by Higman in [Hig].

If $G$ is a Suzuki 2-group, then G has nilpotency class equal to 2, and $G' = Z(G) = \Phi(G) = \Omega( Z(G) ) = \Omega(G)$ and $|G| = |Z(G)|^2$ or $|G| = |Z(G)|^3$. The Suzuki 2-groups $G$ with $|G| = |Z(G)|^2$ are also known as the Suzuki 2-groups of type $A$ and a concrete description of them has been given in [Hig]. For any integer $n \geq 3$ which is not a power of 2 and any odd order automorphism $\theta$ of $\mathbb F_{q}$ where $q = 2^n$ there is such a group of type $A$ denoted by $A(n, \theta)$. And one knows that $A(n,\theta) \cong A(n, \theta')$ if and only if $\theta^{\pm 1} = \theta'$.

I would like to understand the finite 2-groups $P$ whose center $Z(P)$ is of order 2 such that $P/Z(P)$ is a Suzuki 2-group of type $A$. If such a $P$ exists, is it unique?

Moreover, I am happy to impose the following further restrictions on $P$ (if this helps):

- $P$ should have exponent 4
- $\Omega(P)$ is elementary abelian of rank $n+1$ and $P/\Omega(P)$ is elementary abelian of rank $n$.
- $\Omega(P) = \Phi(P) = P' = \mho(P)$

I tried to check some examples with GAP: There is a unique Suzuki 2-group of type $A$ of order 64 and for this group I do know that a group $P$ of order 128 as above exists (and it is unique up to isomorphism). At least the next larger Suzuki 2-groups of type $A$ of order 1024 are capable (i.e. they can be written as $H/Z(H)$ for some group $H$). However, for these one cannot choose $H$ to be a 2-group with $Z(H)$ of order 2.

I would also be interested to know which of the Suzuki 2-groups (not necessarily of type $A$) are capable.

[Hig] Higman, G. "Suzuki 2-groups.", Ill. J. Math. 7, 79-96 (1963).