Example of conjugacy class where both are not real.

Question

In a finite group $G$ a conjugacy class $C$ is called 'real' if $C = C^{-1}$.

Suppose that $H \leq G$ is a subgroup of index $2$.

I am trying to find an examples of a $h \in H$ such that:

• $(h)_H \subsetneq (h)_G$ (equivalently $h$ commutes with no element of $G \setminus H$ ).
• $(h)_H$ is not real.
• $(h)_G$ is not real.

It is easy to find such an example of where $(h)_H$ is not real but $(h)_G$ is, for example any class in $A_n$ that splits a class of $S_n$.

Can anyone provide me with such an example? Or prove that it can never happen. I suspect it can but I can't find an example.

Thanks in advance.

Answer 1

Take $H=(\mathbf{Z}/8\mathbf{Z},+)$, $h=1$, and $G=H\rtimes F$ where $F=\{1,5\}$ viewed as multiplicative group. Then $(h)_H=\{1\}$, $(h)_G=\{1,5\}$; both are non-real.