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Let $G$ be a finite nonabelian group, $\mathbb{Z}G$ its associated integral group ring, and $▵\left(G\right)$ its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups $Q_n\left(G\right)={▵}^n\left(G\right)/{▵}^{n+1}\left(G\right)$ is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.