Let $G$ be a group which is generated by the set of its involutions,
and assume that the product of every two involutions in $G$ has order
a power of 2. Is it possible that $G$ has an element of odd order $\neq 1$?

**Added on Sep 7, 2023:** Dave Benson gave a nice answer for the case that
$G$ is finite. However, the more interesting general case remains open so far.