On the size of $\{a:1\le a<n,n\mathrm{\mid }{a}^2-1,a\mathrm{\mid }{n}^2-1\}$ for number $n$

For number $n>1$, let $\mathcal{A}(n)=\{1\le a<n:n\mathrm{\mid }{a}^2-1,a\mathrm{\mid }{n}^2-1\}$. We show that the size of $\mathcal{A}(n)$ is connected to a property concerning integer evaluations of Fibonacci-like polynomials. In the process, we prove that $\mathrm{\mid }\mathcal{A}(n)\mathrm{\mid }<{\log }_{2}n$, and establish the average value of $\mathrm{\mid }\mathcal{A}(n)\mathrm{\mid }$ to be a little above $2$, asymptotically. But the empirical data up to $n<{10}^7$ indicate that $\mathrm{\mid }\mathcal{A}(n)\mathrm{\mid }\le 3$, proving which is left as an open issue.