  For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the
problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which
the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide
sufficient and necessary conditions and then apply our result to study maximum
scattered linear sets of $\mathrm{PG}(1,q^5)$.
