A Carlitz type result for linearized polynomials

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 $f(x)\mathrm{/}x$ and $g(x)\mathrm{/}x$ coincide. For $n\le 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{P}\mathrm{G}(1,q^5)$.