Permutation polynomials from the trace functions

We study necessary and sufficient conditions on $\gamma$ for several classes of polynomials of the form $X+\gamma {\mathrm{Tr}}_q^{q^n}(h(X))$ to be permutation polynomials over finite field ${\mathbb{F}}_{q^n}$, where $q$ is a prime power, $n$ is a positive integer, and ${\mathrm{Tr}}_q^{q^n}(\cdot )$ denotes the relative trace function from ${\mathbb{F}}_{q^n}$ to ${\mathbb{F}}_q$. In addition, we completely characterize the permutation polynomials of the form $X+\gamma {\mathrm{Tr}}_q^{q^n}(h(X))$ over ${\mathbb{F}}_{q^n}$, with their compositional inverses, where $h(X)=c_{1}X+c_2{X}^2+X^2{\mathrm{Tr}}_q^{q^n}(X),$ $c_1,c_2\in {\mathbb{F}}_q\mathrm{.}$