Comments of ${\mathbb{Z}}_2^2$-supersymmetry in superfield formalism

We investigate superfield formulation of the minimal ${\mathbb{Z}}_2^2$-supersymmetry. It is shown that the integrability on ${\mathbb{Z}}_2^2$-superspace guarantees the invariance of action. Then we present two superfields which carry distinct irreducible representations of the ${\mathbb{Z}}_2^2$-supersymmetry algebra. One of them gives integrable Lagrangian and the other does not. We also show that integrable superfields with different ${\mathbb{Z}}_2^2$-degree also carry irreducible representations and they give invariant actions. To perform this analysis, the representation theory of the minimal ${\mathbb{Z}}_2^2$-supersymmetry algebra is studied in some detail.