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 $\neq 1$?

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.