For a prime p and positive integers m, n, we investigate the compositional inverses of several classes of permutation polynomials of the form \(\sum _{i=1}^kb_i\left( \textrm{Tr}_m^{mn}(x)^{t_i}+\delta \right) ^{s_i}+f_1(x)\) over \( {{\mathbb F}} _{p^{mn}}\) in this paper. Here, for \(1\le i \le k,\) \(s_i\) and \(t_i\) are positive integers, \(b_i \in {{\mathbb F}} _{p^m},\) and \(f_1(x)\) is a polynomial over \( {{\mathbb F}} _{p^{mn}}\) satisfying the following conditions: (i) \(\textrm{Tr}_m^{mn}(x) \circ f_1(x)=\varphi (x) \circ \textrm{Tr}_m^{mn}(x),\) where \(\varphi (x)\) is a polynomial over \( {{\mathbb F}} _{p^m};\) (ii) For any \(a \in {{\mathbb F}} _{p^m},\) \(f_1(x)\) is injective on \(\textrm{Tr}_m^{mn}(a)^{-1}.\)
