I am trying to find subgroups (specifically dihedral subgroups of order $2p^2$) of $\text{Hol}(\mathbb{Z}/2p^2\mathbb{Z})$ (where $p$ is an odd prime), and have found it quite difficult to make progress.
I have that the holomorph is the set of pairs $(a, \rho_b)$ where $\rho_b \in \text{Aut}(\mathbb{Z}/2p^2\mathbb{Z})$ is the function $\rho_b(x) = bx$, for some $b \in (\mathbb{Z}/2p^2\mathbb{Z})^{\times}$.
I have been trying to proceed by finding elements of the holomorph with orders $2$ or $p^2$, but so far without success. Rather than a result, I'd like to know if this approach is likely to yield an answer or if another method would be more likely to work.