Permutation polynomials from the trace functions

We study necessary and sufficient conditions on $\gamma$ for several classes of polynomials of the form $X+\gamma \operatorname{Tr}_{q}^{q^n}(h(X))$ to be permutation polynomials over finite field $\mathbb{F}_{q^n}$, where $q$ is a prime power, $n$ is a positive integer, and $\operatorname{Tr}_{q}^{q^n}(\cdot)$ 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 $X+\gamma\operatorname{Tr}_{q}^{q^n}(h(X))$ over $\mathbb{F}_{q^n}$, with their compositional inverses, where $h(X)=c_1X+c_2X^2+X^2\operatorname{Tr}_{q}^{q^n}(X),$ $c_1,c_2\in\mathbb{F}_q.$