The two extraspecial groups of order $32$ have $20$ and $12$ solutions of $x^2=1$

Given: The objects and hypotheses in the Example.

The generalized dihedral group $\mathrm{Dih}(C_4)$ and the quaternion group $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively).

If $P_1$ and $P_2$ are extraspecial $2$-groups with $t_i$ solutions of $x^2=1$, then $P_1\circ P_2$ has $(t_1{t}_2+(\mathrm{\mid }{P}_1\mathrm{\mid }-t_1)(\mathrm{\mid }{P}_2\mathrm{\mid }-t_2))\mathrm{/}2$ such solutions (A product formula for the number of square roots of the identity in a central product of extraspecial $2$-groups).

$Q_8\circ Q_8\cong \mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$ ($Q_8\circ Q_8\cong \mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$).

For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$ up to isomorphism, with $2^{2n}+2^n$ and $2^{2n}-2^n$ solutions of $x^2=1$ (For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

A set $A$ is finite when $A\approx n$ for some $n\in \mathbb{N}$. (The cardinality $\mid A\mid$ of a finite set).

A central product of extraspecial $2$-groups identified along their centres is extraspecial and has order $\mathrm{\mid }{E}_1\mathrm{\mid }\mathrm{\mid }{E}_2\mathrm{\mid }\mathrm{/}2$ (A central product of extraspecial $p$-groups identified along their centres is extraspecial).