In this paper, we develop the Robinson-Schensted correspondence between the elements of the groups $G_{r}$ $(\mathbb{Z}_{p^{r}}\rtimes \mathbb{Z}^{*}_{p^{r}})$ and $SG_{r}$ $(\mathbb{Z}_{p^{r-1}}\rtimes \mathbb{Z}^{*}_{p^{r}})$, along with a pair of the standard $p$-Young tableaux. This approach differs from the classical method, and ours is based on matrix units arising from orthogonal primitive idempotents computed for every group algebra. Some classical properties of the Robinson-Schensted correspondence are discussed. As a by-product, we also extend the Cauchy identity to our setup, which we refer to as the lacunary Cauchy identity. This study offers new insights into the representation theory of these groups and their combinatorial structures.