Let $G$ be a finite group. Assume that every subgroup of order 4 in $G$ is cyclic (as happens if $G$ is a cyclic group or a generalized quaternion group). It seems to me that it should follow that a Sylow 2-subgroup of $G$ is either a cyclic group or a generalized quaternion group. However, I cannot find a proper reference for this, nor have I been successful on proving it.

I think that this may be well-known, does anyone know a reference I could have a look at?