If $G$ is a finite group, let $\tau(G)$ be equal to probability that a random element $x \in G$ satisfies $x^2=1$, ie. $\tau(G) = \frac{|\{x \in G :\ x^2=1\}|}{|G|}$. Can we say something about $\tau$?
My intuition about the range of possible results we could possibly obtain here comes from another problem: taking two elements of $G$ at random, let $c(G)$ be the probability that they commute. It is quite widely known that if $c(G) > \frac{5}{8}$, then $G$ is abelian and $c(G)=1$. More recently it was proven that the set of possible values of $c(\cdot)$ is nowhere dense, has no irrational limit points, and is well-ordered by $>$.
We can state more or less the same problems in our case:
a) Is there a constant $c<1$ such that $\tau(G)>c$ implies $\tau(G)=1$?
b) Is the set of values of $\tau$ nowhere dense? Does it have any infinite increasing sequence (okay, this is essentially the same as well-order)?
c) Is the set of values of $\tau$ closed?
Parts b, c) look like they might be hard, and I don't expect anyone to write a serious paper in answer to this post, so two more questions, less specific and hopefully easier to answer.
d) Are there any sources of examples of groups (or families of groups) with high value of $\tau$?
Of course $\tau(G)=1$ if and only if $G \simeq \mathbb{Z}_2^k$. We have $\tau(G \times H) = \tau(G) \cdot \tau(H)$, and $\tau(\mathbb{Z}_n) = \frac{1}{n}$ for odd $n$, $\frac{2}{n}$ for even; in particular for abelian groups other than $\mathbb{Z}_2^k$ we have $\tau(G) \leqslant \frac{1}{2}$. We get better efficiency with dihedral groups: we always have $\tau(D_n) > \frac{1}{2}$, although half is the limit with $n \to \infty$. Groups such as $\operatorname{Aut}(D_8)$ also give a half. Therefore it seems like a good bound for a conjecture of the following kind:
e) For every $\varepsilon > 0$ there is an integer $N$, such that $\tau(G) \geqslant \frac{1}{2} + \varepsilon$ and $|G| \geqslant N$ imply $G \simeq H \times \mathbb{Z}_2$ for some group $H$. (As $-\times \mathbb{Z}_2$ does not change $\tau$, this means we have essentially finitely many examples of $G$ giving values bigger than $1/2 + \varepsilon$, and in particular there are finitely many such values.)