Set $\varphi \in \operatorname{Aut}\left( G\right) $ with order $2$. Consider the semidirect product $G\rtimes \left\langle \varphi \right\rangle$. ($G$ is infinite)
Let $\left[ G,\varphi \right] =\left\langle g^{-1}g^\varphi ;g\in G\right\rangle \trianglelefteq G.$
My question is:
Does $\left[ G,\varphi \right] $ have involutions?
I think doesn't have!
Help!