Summary: In this paper, we study the concepts of Wijsman lacunary invariant convergence, Wijsman lacunary invariant statistical convergence, Wijsman lacunary \({\mathcal{I}}_2\)-invariant convergence \(({\mathcal{I}}^{{\sigma}{\theta}}_{W_2} )\), Wijsman lacunary \({\mathcal{I}}^\ast_2\)-invariant convergence \(({\mathcal{I}}^{\ast} {}^{{\sigma}{\theta}}_{W_2} )\), Wijsman \(p\)-strongly lacunary invariant convergence \(([W_2 N_{\sigma \theta}]_p)\) of double sequence of sets and investigate the relationships among Wijsman lacunary invariant convergence, \([W_2 N_{\sigma \theta}]_p, {\mathcal{I}}^{{\sigma}{\theta}}_{W_2}\) and \({\mathcal{I}}^{\ast} {}^{{\sigma}{\theta}}_{W_2} \). Also, we introduce the concepts of \({\mathcal{I}}^{{\sigma}{\theta}}_{W_2} \)-Cauchy double sequence and \({\mathcal{I}}^{\ast} {}^{{\sigma}{\theta}}_{W_2} \)-Cauchy double sequence of sets.