There is a well-known precise formula for the number of involutions of a finite group in terms of character theory, which probably dates back to Frobenius and Schur. For each complex character, $\chi$, of $G$, the Frobenius-Schur indicator of $\chi$, denoted by $\nu(\chi)$, is $\frac{1}{|G|} \sum_{g \in G} \chi(g^{2}).$ We always have $\nu(\chi) \in \{0,-1,1 \}$ and we have $\nu(\chi) = 0 $ if $\chi$ is not real-valued, $\nu(\chi) = 1$ if $\chi$ is afforded by a real representation, $\nu(\chi) = -1$ otherwise.
The number of involutions of $G$ is $\sum_{\chi} \nu(\chi) \chi(1)$, where $\chi$ runs over all the non-trivial irreducible characters of $G$. This simple formula can be powerful. Note that, as expected, this gives no involutions when $G$ has odd order.
Using the Cauchy-Schwarz inequality, we obtain that the number of involutions of $G$ is at most $\sqrt{k-1} \sqrt{|G|-1}$, where $k$ is the number of conjugacy classes of $G$. which plays a role in deriving Brauer-Fowler type estimates.
For example, this gives $[G:C_{G}(t] < \sqrt{k|G|}$ for every involution $t \in G,$ which leads easily to $|G| < k|C_{G}(t)|^{2}$ for each involution $t \in G$ when $G$ has even order. (Later note: This bound is very close to being attained for all the simple groups ${\rm SL}(2,2^{n})$. For ${\rm SL}(2,2^{n})$ has order $(2^{n}-1)2^{n}(2^{n}+1)$ and has $(2^{n}+1)$ conjugacy classes, while each of its involutions has cenralizer of order $2^{n}$).
These facts can be found in many character theory texts, (eg Isaacs).