  In this paper, we study the boomerang spectrum of the power mapping
$F(x)=x^{k(q-1)}$ over ${\mathbb F}_{q^2}$, where $q=p^m$, $p$ is a prime, $m$
is a positive integer and $\gcd(k,q+1)=1$. We first determine the differential
spectrum of $F(x)$ and show that $F(x)$ is locally-APN. This extends a result
of [IEEE Trans. Inf. Theory 57(12):8127-8137, 2011] from $(p,k)=(2,1)$ to
general $(p,k)$. We then determine the boomerang spectrum of $F(x)$ by making
use of its differential spectrum, which shows that the boomerang uniformity of
$F(x)$ is 4 if $p=2$ and $m$ is odd and otherwise it is 2. Our results not only
generalize the results in [Des. Codes Cryptogr. 89:2627-2636, 2021] and
[arXiv:2203.00485, 2022] but also extend the example $x^{45}$ over ${\mathbb
F}_{2^8}$ in [Des. Codes Cryptogr. 89:2627-2636, 2021] into an infinite class
of power mappings with boomerang uniformity 2.
