  In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020]
considered an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the
form $x^3+a(x^{2^s+1})^{2^k}+bx^{3\cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, where
$n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide
new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $ k=0 $, and
$(s,a,b,c)=(m-2,\omega, \omega^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,\omega,
\omega^2,1)$ in which $\omega\in\mathbb{F}_4\setminus \mathbb{F}_2$. By taking
$a=\omega$ and $b=c=\omega^2$, we observe that such kind of quadrinomials can
be rewritten as $a {\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(cx^{2^s+1})$,
where $q=2^m$ and $ {\rm Tr}^n_{m}(x)=x+x^{2^m} $ for $ n=2m$. Inspired by the
quadrinomials and our observation, in this paper we study a class of functions
with the form $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^q{\rm Tr}^{n}_{m}(G(x))$ and
determine the APN-ness of this new kind of functions, where $a \in
\mathbb{F}_{2^n} $ such that $ a+a^q\neq 0$, and both $F$ and $G$ are quadratic
functions over $\mathbb{F}_{2^n}$. We first obtain a characterization of the
conditions for $f(x)$ such that $f(x) $ is an APN function. With the help of
this characterization, we obtain an infinite family of APN functions for $ n=2m
$ with $m$ being an odd positive integer: $ f(x)=a{\rm
Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(b^3x^9) $, where $ a\in \mathbb{F}_{2^n}$
such that $ a+a^q\neq 0 $ and $ b $ is a non-cube in $ \mathbb{F}_{2^n} $.
