  Discrete global symmetries of 4d $\mathcal{N}=2$ SCFTs are studied via two
operations: gauging and twisted compactification. We consider gauging of
discrete symmetries in several well-known 4d $\mathcal{N}=2$ SCFTs, including
$\mathrm{SU}(n)$ SQCD with $2n$ flavors, theories of class $\mathcal{S}$ of
type $A_{2n-1}$, and Argyres--Douglas theories of type $(A_N, A_N)$, as well as
propose new 4d SCFTs as a result. The wreathing technique, which involves
gauging a subgroup of the automorphism group of the quiver diagram of the
corresponding 3d mirror theory, is exploited. This allows us to understand
several properties of discretely gauged theories, including moduli spaces and
how discrete gauging affects the mixed 't Hooft anomaly between the 1-form
symmetry and the 0-form flavor symmetry. Many examples are viewed through the
lens of the Argyres--Seiberg duality and its generalization. We also examine
discrete gauging of $\mathrm{SU}(2)$ SQCD with 4 flavors by various
$\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ subgroups of the
permutation group $S_4$ using the superconformal index. Regarding
compactification, we propose a magnetic quiver for 4d $\mathcal{N}=2$
$\mathrm{SU}(n)$ SQCD with $2n$ flavors compactified on a circle with a
$\mathbb{Z}_2$ twist. The twisted compactification by non-invertible symmetries
of the 4d $\mathcal{N}=4$ SYM theory with gauge group $\mathrm{SU}(N)$ is
revisited. The non-invertible symmetry naturally gives rise to a $\mathbb{Z}_k$
action on the scalar fields parametrizing the moduli space. Upon examining the
$\mathbb{Z}_k$ invariant chiral ring of the Higgs branch, we find that, in
addition to the largest branch of the moduli space that is expected to be
captured by the ABJ(M) theory, there exist in general nilpotent operators that
lead to a branch of the moduli space which is a radical ideal.
