Suppose $G$ is a finite group of order divisible by $8$, with an element $\tau$ of order 2 whose centralizer $C_G(\tau)$ is elementary abelian of order 4. I suspect $G/[G,G]$ must have even order, but I'm not sure how to prove it.
I thought about using transfer and fusion: $\tau$ cannot be conjugate to any involution in the center of a Sylow 2-subgroup (if so, then conjugate that Sylow back so that $\tau$ itself is in the center of a Sylow 2-subgroup, but then that entire Sylow 2-subgroup is contained in the centralizer, contradicting the hypotheses on orders). Since $C_G(\tau)$ contains the center of every Sylow 2-subgroup containing $\tau$ that means that the center of the Sylow 2-subgroups are cyclic of order 2. However, since I don't know much else about the Sylow 2-subgroup, I wasn't sure how to use the transfer.
The goal is to see what sort of classification of groups I can get which have $C_G(\tau)$ of order 4, similar to the one of order 8 mentioned in another question.