In this paper, we investigate several classes of permutation pentanomials over \({{\mathbb {F}}}_{2^{2m}}\) of the form \(f(x)=x^t+x^{r_1(q-1)+t}+x^{r_2(q-1)+t}+x^{r_3(q-1)+t}+x^{r_4(q-1)+t}\) with \( 1\le r_i\le t\) for \(i\in [1,4]\) . A new technique is presented to describe the sufficient condition for f(x) to be a permutation through investigating two kinds of irreducible factors, which are called polynomials of nonzero trace and zero trace, of some certain polynomials over \({{\mathbb {F}}}_{2}\) . We resolve the open problem the authors left in Zhang et al. (Finite Fields Appl 98:102468, 2024). Numerical results suggest that the results in this paper seem to contain all the permutation pentanomials of that form with \(\textrm{gcd}(x^{r_4}+x^{r_3}+x^{r_2}+x^{r_1}+1,x^t+x^{t-r_1}+x^{t-r_2}+x^{t-r_3}+x^{t-r_4})=1\) for \(t>23\) and the conditions presented in Theorems 3.1, 3.7, 3.10 and 3.14 of this paper are also necessary.
