Unfortunately the last part isn’t true since $C_p\times C_p$ is metacyclic. Did you have something else in mind?

@MikhailBorovoi Yes $\text{tr}$ is short for transfer also known as restriction.

@MikhailBorovoi I replaced the original incorrect argument by a much simpler one.

@WillSawin Aah, I missed that you only looked at the set of elements having a fixed point, not the subgroup generated by this set.... :-( Sorry!

@WillSawin I think the answer to your question is no (or I might have misunderstood something): The dihedral group $D_8$ acts transitively on the set of left cosets $D_8/\langle r\rangle$, where $r$ is a reflection. The set of elements having a fixed point has index $2$, but $D_8/\langle r\rangle$ has $4$ elements.