Prime injections and quasipolarities

Let $p$ be a prime number. Consider the injection $$\iota :\mathbb{Z}\mathrm{/}n\mathbb{Z}\to \mathbb{Z}\mathrm{/}pn\mathbb{Z}:x\mapsto px,$$ and the elements $e^{u}.v:=(u,v)\in $\mathbb{Z}$/n$\mathbb{Z}$\rtimes $\mathbb{Z}$/n$\mathbb{Z}$^{\times}$and$e^{w}.r:=(w,r)\in $\mathbb{Z}${p n}\rtimes $\mathbb{Z}${p n}^{\times}$\mathrm{.}Suppose$e^{u}.v\in $\mathbb{Z}$/n$\mathbb{Z}$\rtimes $\mathbb{Z}$/n$\mathbb{Z}$^{\times}$isseenasanautomorphismof$$\mathbb{Z}$/n$\mathbb{Z}$$by$e^{u}.v(x)=vx+u$;then$e^{u}.v$ is a quasipolarity if it is an involution without fixed points. In this brief note give an explicit formula for the number of quasipolarites of $\mathbb{Z}\mathrm{/}n\mathbb{Z}$ in terms of the prime decomposition of $n$, and we prove sufficient conditions such that $(e^{w}.r)\circ \iota =\iota\circ (e^{u}.v)$,where$e^{w}.r$and$e^{u}.v$ are quasipolarities.