  Irreducible representations (irreps) of $\mathbb{Z}_2^2$-graded supersymmetry
algebra of ${\cal N}=2$ are obtained by the method of induced representation
and they are used to derive $\mathbb{Z}_2^2$-graded supersymmetric classical
actions. The irreps are four dimensional for $ \lambda = 0$ where $ \lambda $
is an eigenvalue of the Casimir element, and eight dimensional for $\lambda
\neq 0.$ The eight dimensional irreps reduce to four dimensional ones only when
$\lambda$ and an eigenvalue of Hamiltonian satisfy a particular relation. The
reduced four dimensional irreps are used to define $\mathbb{Z}_2^2$-graded
supersymmetry transformations and two types of classical actions invariant
under the transformations are presented. It is shown that one of the Noether
charges vanishes if all the variables of specific $\mathbb{Z}_2^2$-degree are
auxiliary.
