Permutation polynomials (PPs) of the form (xq-x + c)q2-1/3+1 + x over Fq2 were presented by Li et al.(Finite Fields Appl 22:16-23,2013).More recently,we have constructed PPs of the form (xq + bx + c)q2-1/d+1-bx over Fq2,where d =2,3,4,6 (Yuan and Zheng in Finite Fields Appl 35:215-230,2015).In this paper we concentrate our efforts on the PPs of more general form f (x) =(axq + bx + c)r φ((axq + bx + c)(q2-1)/d)+ uxq + vx over Fq2,where a,b,c,u,v ∈ Fq2,r ∈ Z+,φ(x) ∈ Fq2[x]and d is an arbitrary positive divisor of q2-1.The key step is the construction of a commutative diagram with specific properties,which is the basis of the Akbary-Ghioca-Wang (AGW) criterion.By employing the AGW criterion two times,we reduce the problem of determining whether f (x) permutes Fq2 to that of verifying whether two more polynomials permute two subsets of Fq2.AS a consequence,we find a series of simple conditions for f(x) to be a PP of Fq2.These results unify and generalize some known classes of PPs.