Fixed points of the sum of divisors function on \({{\mathbb{F}}}_2[x]\)

We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points \(F\) sum divisors function \(\sigma : {\mathbb{F}}_2[x] \mapsto {\mathbb{F}}_2[x]\) (defined mutatis mutandi like usual integers) form \(F := A^2 \cdot S\), \(S\) square-free, with \(\omega(S) \leq 3\), coprime \(A\), for \(A\) even, whatever degree, under some conditions. This gives characterization \(5\) \(11\) known \(\sigma\) in \({\mathbb{F}}_2[x]\).